Mathletic | Practice Platform

AMC 10·2000 · AMC 10A·Medium

Two different prime numbers between

4

and

18

are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?

AMC 10·2000 · AMC 10A·Easy

The Fibonacci sequence

1,1,2,3,5,8,13,21,\ldots

starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

AMC 10·2001 · AMC 10A·Medium

Suppose that

n

is the product of three consecutive integers and that

n

is divisible by

7

. Which of the following is not necessarily a divisor of

n

?

AMC 10·2001 · AMC 10A·Medium

A charity sells

140

benefit tickets for a total of

\$2001

. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?

AMC 10·2001 · AMC 10A·Easy

Let

P(n)

and

S(n)

denote the product and the sum, respectively, of the digits of the integer

n

. For example,

P(23) = 6

and

S(23) = 5

. Suppose

N

is a two-digit number such that

N = P(N)+S(N)

. What is the units digit of

N

?

AMC 10·2001 · AMC 10A·Easy

Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?

AMC 10·2002 · AMC 10A·Medium

Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?

AMC 10·2002 · AMC 10A·Hard

A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square , and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?

AMC 10·2002 · AMC 10B·Medium

The number

25^{64}\cdot 64^{25}

is the square of a positive integer

N

. In decimal representation, the sum of the digits of

N

is

AMC 10·2002 · AMC 10B·Medium

The positive integers

A, B, A-B,

and

A+B

are all prime numbers. The sum of these four primes is

AMC 10·2002 · AMC 10B·Easy

For how many positive integers

n

is

n^2 - 3n + 2

a prime number?

AMC 10·2002 · AMC 10 Practice·Medium

For how many positive integers

n

is

n^3 - 8n^2 + 20n - 13

a prime number?

AMC 10·2002 · AMC 10 Practice·Easy

Which of the following numbers is a perfect square?

AMC 10·2002 · AMC 10 Practice·Easy

How many ordered triples of positive integers

(x,y,z)

satisfy

(x^y)^z=64?

AMC 10·2003 · AMC 10A·Medium

Let

n

be the largest integer that is the product of exactly 3 distinct prime numbers

d

,

e

, and

10d+e

, where

d

and

e

are single digits. What is the sum of the digits of

n

?

AMC 10·2003 · AMC 10A·Medium

What is the units digit of

13^{2003}

?

AMC 10·2003 · AMC 10A·Hard

Sally has five red cards numbered

1

through

5

and four blue cards numbered

3

through

6

. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

AMC 10·2003 · AMC 10A·Hard

Let

n

be a

5

-digit number, and let

q

and

r

be the quotient and the remainder, respectively, when

n

is divided by

100

. For how many values of

n

is

q+r

divisible by

11

?

AMC 10·2003 · AMC 10B·Medium

Let

\clubsuit(x)

denote the sum of the digits of the positive integer

x

. For example,

\clubsuit(8)=8

and

\clubsuit(123)=1+2+3=6

. For how many two-digit values of

x

is

\clubsuit(\clubsuit(x))=3

?

AMC 10·2003 · AMC 10B·Medium

There are

100

players in a single tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest

28

players are given a bye, and the remaining

72

players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is

AMC 10·2003 · AMC 10B·Hard

How many distinct four-digit numbers are divisible by

3

and have

23

as their last two digits?

AMC 10·2004 · AMC 10B·Easy

How many two-digit positive integers have at least one

7

as a digit?

AMC 10·2004 · AMC 10B·Hard

Let

1

;

4

;

7

;

\ldots

and

9

;

16

;

23

;

\ldots

be two arithmetic progressions. The set

S

is the union of the first

2004

terms of each sequence. How many distinct numbers are in

S

?

AMC 10·2004 · AMC 10B·Easy

A standard six-sided die is rolled, and

P

is the product of the five numbers that are visible. What is the largest number that is certain to divide

P

?

AMC 10·2004 · AMC 10B·Easy

Which of the following numbers is a perfect square?

AMC 10·2004 · AMC 10B·Easy

On a trip from the United States to Canada, Isabella took

d

U.S. dollars. At the border she exchanged them all, receiving

10

Canadian dollars for every

7

U.S. dollars. After spending

60

Canadian dollars, she had

d

Canadian dollars left. What is the sum of the digits of

d

?

AMC 10·2005 · AMC 10A·Medium

How many positive cubes divide

3! \cdot 5! \cdot 7!

?

AMC 10·2005 · AMC 10A·Medium

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is

6

. How many two-digit numbers have this property?

AMC 10·2005 · AMC 10A·Hard

Let

S

be the set of the

2005

smallest positive multiples of

4

, and let

T

be the set of the

2005

smallest positive multiples of

6

. How many elements are common to

S

and

T

?

AMC 10·2005 · AMC 10A·Hard

For each positive integer

n > 1

, let

P(n)

denote the greatest prime factor of

n

. For how many positive integers

n

is it true that both

P(n) = \sqrt{n}

and

P(n+48) = \sqrt{n+48}

?

AMC 10·2005 · AMC 10B·Medium

The first term of a sequence is

2005

. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the

{2005}^{\text{th}}

term of the sequence?

AMC 10·2005 · AMC 10B·Hard

For how many positive integers

n

less than or equal to

24

is

n!

evenly divisible by

1 + 2 + \cdots + n?

AMC 10·2005 · AMC 10B·Hard

Let

x

and

y

be two-digit integers such that

y

is obtained by reversing the digits of

x

. The integers

x

and

y

satisfy

x^2 - y^2 = m^2

for some positive integer

m

. What is

x + y + m

?

AMC 10·2006 · AMC 10A·Easy

For how many real values of

x

is

\sqrt{120-\sqrt{x}}

an integer?

AMC 10·2006 · AMC 10A·Hard

How many four-digit positive integers have at least one digit that is a

2

or a

3

?

AMC 10·2006 · AMC 10A·Hard

Two farmers agree that pigs are worth

300

dollars and that goats are worth

210

dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a

390

dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

AMC 10·2006 · AMC 10B·Medium

What is the tens digit in the sum

7!+8!+9!+...+2006!

AMC 10·2006 · AMC 10B·Medium

Leap Day, February

29

,

2004

, occurred on a Sunday. On what day of the week will Leap Day, February

29

,

2020

, occur?

AMC 10·2006 · AMC 10B·Medium

Let

a_1 , a_2 , ...

be a sequence for which

a_1=2

,

a_2=3

, and

a_n=\frac{a_{n-1}}{a_{n-2}}

for each positive integer

n \ge 3

. What is

a_{2006}

?

AMC 10·2006 · AMC 10B·Hard

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?

AMC 10·2007 · AMC 10A·Hard

A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let

S

be the sum of all the terms in the sequence. What is the largest prime factor that always divides

S

?

AMC 10·2007 · AMC 10B·Hard

Let

n

denote the smallest positive integer that is divisible by both

4

and

9,

and whose base-

10

representation consists of only

4

's and

9

's, with at least one of each. What are the last four digits of

n?

AMC 10·2008 · AMC 10A·Hard

Let

k={2008}^{2}+{2}^{2008}

. What is the units digit of

k^2+2^k

?

AMC 10·2009 · AMC 10A·Easy

What is the sum of the digits of the square of

\text 111111111

?

AMC 10·2010 · AMC 10A·Easy

Marvin had a birthday on Tuesday, May 27 in the leap year

2008

. In what year will his birthday next fall on a Saturday?

AMC 10·2010 · AMC 10A·Hard

The number obtained from the last two nonzero digits of

90!

is equal to

n

. What is

n

?

AMC 10·2010 · AMC 10A·Hard

Jim starts with a positive integer

n

and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with

n = 55

, then his sequence contains

5

numbers:

\begin{array}{ccccc} {}&{}&{}&{}&55\\ 55&-&7^2&=&6\\ 6&-&2^2&=&2\\ 2&-&1^2&=&1\\ 1&-&1^2&=&0\\ \end{array}

Let

N

be the smallest number for which Jim’s sequence has

8

numbers. What is the units digit of

N

?

AMC 10·2010 · AMC 10A·Easy

A

\text{palindrome}

, such as

83438

, is a number that remains the same when its digits are reversed. The numbers

x

and

x+32

are three-digit and four-digit palindromes, respectively. What is the sum of the digits of

x

?

AMC 10·2010 · AMC 10B·Easy

A month with

31

days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

AMC 10·2011 · AMC 10A·Easy

A majority of the

30

students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than

1

. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was

\$17.71

. What was the cost of a pencil in cents?

AMC 10·2011 · AMC 10A·Medium

How many even integers are there between

200

and

700

whose digits are all different and come from the set

\left\{1,2,5,7,8,9\right\}

?

AMC 10·2011 · AMC 10A·Medium

In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?

AMC 10·2011 · AMC 10A·Hard

Seven students count from 1 to 1000 as follows: Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000. Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers. Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers. Finally, George says the only number that no one else says. What number does George say?

AMC 10·2011 · AMC 10B·Hard

What is the hundreds digit of

2011^{2011}?

AMC 10·2011 · AMC 10B·Easy

In multiplying two positive integers

a

and

b

, Ron reversed the digits of the two-digit number

a

. His erroneous product was

161

. What is the correct value of the product of

a

and

b

?

AMC 10·2012 · AMC 10A·Medium

A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?

AMC 10·2012 · AMC 10A·Medium

Chubby makes nonstandard checkerboards that have

31

squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?

AMC 10·2012 · AMC 10A·Medium

Let

a

and

b

be relatively prime positive integers with

a>b>0

and

\dfrac{a^3-b^3}{(a-b)^3} = \dfrac{73}{3}

. What is

a-b

?

AMC 10·2012 · AMC 10B·Easy

When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover?

AMC 10·2013 · AMC 10A·Medium

How many three-digit numbers are not divisible by

5

, have digits that sum to less than

20

, and have the first digit equal to the third digit?

AMC 10·2013 · AMC 10A·Medium

Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next

365

-day period will exactly two friends visit her?

AMC 10·2013 · AMC 10A·Easy

A softball team played ten games, scoring

1

,

2

,

3

,

4

,

5

,

6

,

7

,

8

,

9

, and

10

runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

AMC 10·2013 · AMC 10B·Medium

The number

2013

has the property that its units digit is the sum of its other digits, that is

2+0+1=3

. How many integers less than

2013

but greater than

1000

have this property?

AMC 10·2013 · AMC 10B·Hard

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is

N

. What is the smallest possible value of

N

?

AMC 10·2013 · AMC 10B·Hard

A positive integer

n

is nice if there is a positive integer

m

with exactly four positive divisors (including

1

and

m

) such that the sum of the four divisors is equal to

n

. How many numbers in the set

\{ 2010,2011,2012,\dotsc,2019 \}

are nice?

AMC 10·2013 · AMC 10B·Hard

Bernardo chooses a three-digit positive integer

N

and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer

S

. For example, if

N = 749

, Bernardo writes the numbers

10,\!444

and

3,\!245

, and LeRoy obtains the sum

S = 13,\!689

. For how many choices of

N

are the two rightmost digits of

S

, in order, the same as those of

2N

?

AMC 10·2013 · AMC 10B·Easy

Three positive integers are each greater than

1

, have a product of

27000

, and are pairwise relatively prime. What is their sum?

AMC 10·2014 · AMC 10A·Easy

Which of the following numbers is a perfect square?

AMC 10·2015 · AMC 10A·Medium

Consider the set of all fractions

\frac{x}{y}

, where

x

and

y

are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by

1

, the value of the fraction is increased by

10\%

?

AMC 10·2015 · AMC 10A·Medium

Hexadecimal (base-16) numbers are written using numeric digits

0

through

9

as well as the letters

A

through

F

to represent

10

through

15

. Among the first

1000

positive integers, there are

n

whose hexadecimal representation contains only numeric digits. What is the sum of the digits of

n

?

AMC 10·2015 · AMC 10A·Medium

A rectangle with positive integer side lengths in

\text{cm}

has area

A \text{cm}^2

and perimeter

P \text{cm}

. Which of the following numbers cannot equal

A+P

?

AMC 10·2015 · AMC 10B·Medium

Among the positive integers less than

100

, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?

AMC 10·2015 · AMC 10B·Hard

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than

5

steps left). Suppose Dash takes

19

fewer jumps than Cozy to reach the top of the staircase. Let

s

denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of

s

?

AMC 10·2015 · AMC 10B·Hard

Let

n

be a positive integer greater than

4

such that the decimal representation of

n!

ends in

k

zeros and the decimal representation of

(2n)!

ends in

3k

zeros. Let

s

denote the sum of the four least possible values of

n

. What is the sum of the digits of

s

?

AMC 10·2015 · AMC 10B·Easy

Four siblings ordered an extra large pizza. Alex ate

\frac15

, Beth

\frac13

, and Cyril

\frac14

of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?

AMC 10·2016 · AMC 10A·Hard

For some positive integer

n

, the number

110n^3

has

110

positive integer divisors, including

1

and the number

110n^3

. How many positive integer divisors does the number

81n^4

have?

AMC 10·2016 · AMC 10A·Hard

A binary operation

\diamondsuit

has the properties that

a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c

and that

a\,\diamondsuit \,a=1

for all nonzero real numbers

a, b,

and

c

. (Here

\cdot

represents multiplication). The solution to the equation

2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100

can be written as

\tfrac{p}{q}

, where

p

and

q

are relatively prime positive integers. What is

p+q?

AMC 10·2016 · AMC 10A·Hard

How many ordered triples

(x,y,z)

of positive integers satisfy

\operatorname{lcm}(x,y) = 72, \operatorname{lcm}(x,z) = 600 \text{ and \operatorname{lcm}}(y,z)=900

?

AMC 10·2016 · AMC 10A·Easy

Star lists the whole numbers

1

through

30

once. Emilio copies Star's numbers, replacing each occurrence of the digit

2

by the digit

1

. Star adds her numbers and Emilio adds his numbers. How much larger is Star's sum than Emilio's?

AMC 10·2016 · AMC 10A·Easy

A triangular array of

2016

coins has

1

coin in the first row,

2

coins in the second row,

3

coins in the third row, and so on up to

N

coins in the

N

th row. What is the sum of the digits of

N

?

AMC 10·2016 · AMC 10B·Easy

Zoey read

15

books, one at a time. The first book took her

1

day to read, the second book took her

2

days to read, the third book took her

3

days to read, and so on, with each book taking her

1

more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day the week did she finish her

15

th book?

AMC 10·2016 · AMC 10B·Easy

Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number

S

. What is the smallest possible value for the sum of the digits of

S

?

AMC 10·2016 · AMC 10B·Easy

What is the tens digit of

2015^{2016}-2017?

AMC 10·2017 · AMC 10A·Medium

Define a sequence recursively by

F_{0}=0, F_{1}=1,

and

F_{n}=

the remainder when

F_{n-1}+F_{n-2}

is divided by

3,

for all

n\geq 2.

Thus the sequence starts

0,1,1,2,0,2,\ldots

What is

F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}?

AMC 10·2017 · AMC 10A·Medium

There are

10

horses, named Horse

1

, Horse

2

, . . . , Horse

10

. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse

k

runs one lap in exactly

k

minutes. At time

0

all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time

S > 0

, in minutes, at which all

10

horses will again simultaneously be at the starting point is

S=2520

. Let

T > 0

be the least time, in minutes, such that at least

5

of the horses are again at the starting point. What is the sum of the digits of

T?

AMC 10·2017 · AMC 10A·Medium

Let

S(n)

equal the sum of the digits of positive integer

n

. For example,

S(1507) = 13

. For a particular positive integer

n

,

S(n) = 1274

. Which of the following could be the value of

S(n+1)

?

AMC 10·2017 · AMC 10B·Easy

Mary thought of a positive two-digit number. She multiplied it by

3

and added

11

. Then she switched the digits of the result, obtaining a number between

71

and

75

, inclusive. What was Mary's number?

AMC 10·2017 · AMC 10B·Medium

How many of the base-ten numerals for the positive integers less than or equal to

2017

contain the digit

0

?

AMC 10·2017 · AMC 10B·Medium

Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example,

3

,

23578

, and

987620

are monotonous, but

88

,

7434

, and

23557

are not. How many monotonous positive integers are there?

AMC 10·2017 · AMC 10B·Hard

Let

N=123456789101112\dots4344

be the

79

-digit number that is formed by writing the integers from

1

to

44

in order, one after the other. What is the remainder when

N

is divided by

45

?

AMC 10·2018 · AMC 10A·Medium

A number

m

is randomly selected from the set

\{11,13,15,17,19\}

, and a number

n

is randomly selected from

\{1999,2000,2001,\ldots,2018\}

. What is the probability that

m^n

has a units digit of

1

?

AMC 10·2018 · AMC 10A·Hard

Let

a, b, c,

and

d

be positive integers such that

\gcd(a, b)=24

,

\gcd(b, c)=36

,

\gcd(c, d)=54

, and

70<\gcd(d, a)<100

. Which of the following must be a divisor of

a

?

AMC 10·2018 · AMC 10A·Hard

For a positive integer

n

and nonzero digits

a

,

b

, and

c

, let

A_n

be the

n

-digit integer each of whose digits is equal to

a

; let

B_n

be the

n

-digit integer each of whose digits is equal to

b

, and let

C_n

be the

2n

-digit (not

n

-digit) integer each of whose digits is equal to

c

. What is the greatest possible value of

a + b + c

for which there are at least two values of

n

such that

C_n - B_n = A_n^2

?

AMC 10·2018 · AMC 10B·Medium

Which of the following expressions is never a prime number when

p

is a prime number?

AMC 10·2018 · AMC 10B·Medium

How many of the first

2018

numbers in the sequence

101, 1001, 10001, 100001, \dots

are divisible by

101

?

AMC 10·2018 · AMC 10B·Medium

Let

a_1,a_2,\dots,a_{2018}

be a strictly increasing sequence of positive integers such that

a_1+a_2+\cdots+a_{2018}=2018^{2018}.

What is the remainder when

a_1^3+a_2^3+\cdots+a_{2018}^3

is divided by

6

?

AMC 10·2018 · AMC 10B·Medium

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is

1

year older than Chloe, and Zoe is exactly

1

year old today. Today is the first of the

9

birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

AMC 10·2018 · AMC 10B·Hard

Mary chose an even

4

-digit number

n

. She wrote down all the divisors of

n

in increasing order from left to right:

1,2,\ldots,\dfrac{n}{2},n

. At some moment Mary wrote

323

as a divisor of

n

. What is the smallest possible value of the next divisor written to the right of

323

?

AMC 10·2018 · AMC 10B·Hard

How many ordered pairs

(a, b)

of positive integers satisfy the equation

a\cdot b + 63 = 20\cdot \operatorname{lcm}(a, b) + 12\cdot\gcd(a,b),

where

\gcd(a,b)

denotes the greatest common divisor of

a

and

b

, and

\operatorname{lcm}(a,b)

denotes their least common multiple?

AMC 10·2019 · AMC 10A·Easy

A rectangular floor that is

10

feet wide and

17

feet long is tiled with

170

one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?

AMC 10·2019 · AMC 10A·Medium

How many positive integer divisors of

201^9

are perfect squares or perfect cubes (or both)?

AMC 10·2019 · AMC 10A·Medium

A sequence of numbers is defined recursively by

a_1 = 1

,

a_2 = \frac{3}{7}

, and

a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}

for all

n \geq 3

Then

a_{2019}

can be written as

\frac{p}{q}

, where

p

and

q

are relatively prime positive integers. What is

p+q ?

AMC 10·2019 · AMC 10A·Easy

What is the hundreds digit of

(20!-15!)?

AMC 10·2019 · AMC 10A·Easy

What is the greatest three-digit positive integer

n

for which the sum of the first

n

positive integers is

\underline{not}

a divisor of the product of the first

n

positive integers?

AMC 10·2019 · AMC 10B·Medium

The base-ten representation for

19!

is

121,6T5,100,40M,832,H00

, where

T

,

M

, and

H

denote digits that are not given. What is

T+M+H

?

AMC 10·2019 · AMC 10B·Medium

Let

S

be the set of all positive integer divisors of

100,000.

How many numbers are the product of two distinct elements of

S?

AMC 10·2019 · AMC 10B·Easy

Consider the statement, "If

n

is not prime, then

n-2

is prime." Which of the following values of

n

is a counterexample to this statement?

AMC 10·2019 · AMC 10B·Easy

There is a positive integer

n

such that

(n+1)! + (n+2)! = n! \cdot 440

. What is the sum of the digits of

n

?

AMC 10·2019 · AMC 10B·Easy

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either

12

pieces of red candy,

14

pieces of green candy,

15

pieces of blue candy, or

n

pieces of purple candy. A piece of purple candy costs

20

cents. What is the smallest possible value of

n

?

AMC 10·2020 · AMC 10A·Medium

A positive integer divisor of

12!

is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

?

AMC 10·2020 · AMC 10A·Hard

For how many positive integers

n \le 1000

is

\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor

not divisible by

3

? (Recall that

\lfloor x \rfloor

is the greatest integer less than or equal to

x

.)

AMC 10·2020 · AMC 10A·Hard

Let

n

be the least positive integer greater than

1000

for which

\gcd(63, n+120) =21 \text{and} \gcd(n+63, 120)=60.

What is the sum of the digits of

n

?

AMC 10·2020 · AMC 10A·Easy

How many

4

-digit positive integers (that is, integers between

1000

and

9999

, inclusive) having only even digits are divisible by

5?

AMC 10·2020 · AMC 10A·Easy

A single bench section at a school event can hold either

7

adults or

11

children. When

N

bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of

N?

AMC 10·2020 · AMC 10B·Hard

What is the remainder when

2^{202} +202

is divided by

2^{101}+2^{51}+1

?

AMC 10·2020 · AMC 10B·Easy

Driving along a highway, Megan noticed that her odometer showed

15951

(miles). This number is a palindrome-it reads the same forward and backward. Then

2

hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this

2

-hour period?

AMC 10·2021 · AMC 10A·Medium

For which of the following integers

b

is the base-

b

number

2021_b - 221_b

not divisible by

3

?

AMC 10·2021 · AMC 10A·Medium

Let

f

be a function defined on the set of positive rational numbers with the property that

f(a\cdot b)=f(a)+f(b)

for all positive rational numbers

a

and

b

. Suppose that

f

also has the property that

f(p)=p

for every prime number

p

. For which of the following numbers

x

is

f(x)<0

?

AMC 10·2021 · AMC 10A·Easy

The sum of two natural numbers is

17{,}402

. One of the two numbers is divisible by

10

. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

AMC 10·2021 · AMC 10B·Medium

Let

N = 34 \cdot 34 \cdot 63 \cdot 270

. What is the ratio of the sum of the odd divisors of

N

to the sum of the even divisors of

N

?

AMC 10·2021 · AMC 10B·Medium

Let

n

be a positive integer and

d

be a digit such that the value of the numeral

\underline{32d}

in base

n

equals

263

, and the value of the numeral

\underline{324}

in base

n

equals the value of the numeral

\underline{11d1}

in base six. What is

n + d ?

AMC 10·2021 · AMC 10B·Medium

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example,

1357, 89,

and

5

are all uphill integers, but

32, 1240,

and

466

are not. How many uphill integers are divisible by

15

?

AMC 10·2021 · AMC 10B·Hard

Ang, Ben, and Jasmin each have

5

blocks, colored red, blue, yellow, white, and green; and there are

5

empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives

3

blocks all of the same color is

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. What is

m + n ?

AMC 10·2021 · AMC 10B·Easy

The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give

24

, while the other two multiply to

30

. What is the sum of the ages of Jonie's four cousins?

AMC 10·2021 · AMC 10A Fall·Medium

The base-nine representation of the number

N

is

27{,}006{,}000{,}052_{\text{nine}}.

What is the remainder when

N

is divided by

5?

AMC 10·2021 · AMC 10A Fall·Hard

For each positive integer

n

, let

f_1(n)

be twice the number of positive integer divisors of

n

, and for

j \ge 2

, let

f_j(n) = f_1(f_{j-1}(n))

. For how many values of

n \le 50

is

f_{50}(n) = 12?

AMC 10·2021 · AMC 10A Fall·Easy

The six-digit number

\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}

is prime for only one digit

A.

What is

A?

AMC 10·2021 · AMC 10A Fall·Easy

A two-digit positive integer is said to be

\emph{cuddly}

if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?

AMC 10·2021 · AMC 10B Fall·Easy

Forty slips of paper numbered

1

to

40

are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by

100

and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?

AMC 10·2021 · AMC 10B Fall·Hard

For each integer

n\geq 2

, let

S_n

be the sum of all products

jk

, where

j

and

k

are integers and

1\leq j<k\leq n

. What is the sum of the 10 least values of

n

such that

S_n

is divisible by

3

?

AMC 10·2021 · AMC 10B Fall·Easy

The greatest prime number that is a divisor of

16{,}384

is

2

because

16{,}384 = 2^{14}

. What is the sum of the digits of the greatest prime number that is a divisor of

16{,}383

?

AMC 10·2022 · AMC 10A·Medium

Define

L_n

as the least common multiple of all the integers from

1

to

n

inclusive. There is a unique integer

h

such that

\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{17}=\frac{h}{L_{17}}

What is the remainder when

h

is divided by

17

?

AMC 10·2022 · AMC 10B·Medium

The positive difference between a pair of primes is equal to

2

, and the positive difference between the cubes of the two primes is

31106

. What is the sum of the digits of the least prime that is greater than those two primes?

AMC 10·2022 · AMC 10B·Medium

One of the following numbers is not divisible by any prime number less than

10.

Which is it?

AMC 10·2022 · AMC 10B·Easy

How many three-digit positive integers have an odd number of even digits?

AMC 10·2022 · AMC 10B·Easy

How many of the first ten numbers of the sequence

121, 11211, 1112111, \ldots

are prime numbers?

AMC 10·2023 · AMC 10A·Medium

In a table tennis tournament, every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was

40\%

more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

AMC 10·2023 · AMC 10A·Hard

If the positive integer

n

has positive integer divisors

a

and

b

with

n = ab

, then

a

and

b

are said to be

\textit{complementary}

divisors of

n

. Suppose that

N

is a positive integer that has one complementary pair of divisors that differ by

20

and another pair of complementary divisors that differ by

23

. What is the sum of the digits of

N

?

AMC 10·2023 · AMC 10A·Easy

How many positive perfect squares less than

2023

are divisible by

5

?

AMC 10·2023 · AMC 10A·Easy

How many digits are in the base-ten representation of

8^5 \cdot 5^{10} \cdot 15^2

?

AMC 10·2023 · AMC 10B·Medium

Suppose

a

,

b

, and

c

are positive integers such that

\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.

Which of the following statements are necessarily true? I. If

\gcd(a,14)=1

or

\gcd(b,15)=1

or both, then

\gcd(c,210)=1

. II. If

\gcd(c,210)=1

, then

\gcd(a,14)=1

or

\gcd(b,15)=1

or both. III.

\gcd(c,210)=1

if and only if

\gcd(a,14)=\gcd(b,15)=1

.

AMC 10·2023 · AMC 10B·Easy

What is the units digit of

2022^{2023} + 2023^{2022}

?

AMC 10·2023 · AMC 10B·Easy

The numbers

16

and

25

are a pair of consecutive positive squares whose difference is

9

. How many pairs of consecutive positive perfect squares have a difference of less than or equal to

2023

?

AMC 10·2024 · AMC 10A·Easy

Consider the following operation. Given a positive integer

n

, if

n

is a multiple of

3

, then you replace

n

by

\frac{n}{3}

. If

n

is not a multiple of

3

, then you replace

n

by

n+10

. Then continue this process. For example, beginning with

n=4

, this procedure gives

4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to \cdots

. Suppose you start with

n=100

. What value results if you perform this operation exactly

100

times?

AMC 10·2024 · AMC 10A·Medium

Let

M

be the greatest integer such that both

M+1213

and

M+3773

are perfect squares. What is the units digit of

M

?

AMC 10·2024 · AMC 10A·Medium

There are exactly

K

positive integers

b

with

5 \leq b \leq 2024

such that the base-

b

integer

2024_b

is divisible by

16

(where

16

is in base ten). What is the sum of the digits of

K

?

AMC 10·2024 · AMC 10A·Easy

What is the sum of the digits of the smallest prime that can be written as a sum of

5

distinct primes?

AMC 10·2024 · AMC 10A·Easy

What is the least value of

n

such that

n!

is a multiple of

2024

?

AMC 10·2024 · AMC 10B·Hard

Let

P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}

How many of the values

P(2022)

,

P(2023)

,

P(2024)

, and

P(2025)

are integers?

AMC 10·2024 · AMC 10B·Hard

Each of

27

bricks (right rectangular prisms) has dimensions

a \times b \times c

, where

a

,

b

, and

c

are pairwise relatively prime positive integers. These bricks are arranged to form a

3 \times 3 \times 3

block, as shown on the left below. A

28

th brick with the same dimensions is introduced, and these bricks are reconfigured into a

2 \times 2 \times 7

block, shown on the right. The new block is

1

unit taller,

1

unit wider, and

1

unit deeper than the old one. What is

a + b + c

?

AMC 10·2024 · AMC 10B·Easy

What is the remainder when

7^{2024}+7^{2025}+7^{2026}

is divided by

19

?

AMC 10·2024 · AMC 10B·Easy

Let

N

be the product of all the positive integer divisors of

42

. What is the units digit of

N

?

Number Theory