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Number Theory set

20 problems, difficulty 2–4, one per topic where possible

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Number Theory setAMC 10 Step by Step

Name: Date: September 13, 2026

20 problems · 60 minutes · 6 / 1.5 / 0

1.2014 AMC 10A #8
Which of the following numbers is a perfect square?
1. A)
  $\dfrac{14!15!}2$
2. B)
  $\dfrac{15!16!}2$
3. C)
  $\dfrac{16!17!}2$
4. D)
  $\dfrac{17!18!}2$
5. E)
  $\dfrac{18!19!}2$
2.2015 AMC 10B #11
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?
1. A)
  $\dfrac{8}{99}$
2. B)
  $\dfrac{2}{5}$
3. C)
  $\dfrac{9}{20}$
4. D)
  $\dfrac{1}{2}$
5. E)
  $\dfrac{9}{16}$
3.2021 AMC 10A #8
When a student multiplied the number $66$ by the repeating decimal, $\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},$ where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$ -digit number $\underline{a} \ \underline{b}?$
1. A)
  $15$
2. B)
  $30$
3. C)
  $45$
4. D)
  $60$
5. E)
  $75$
4.2024 AMC 10A #3
What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes?
1. A)
  $5$
2. B)
  $7$
3. C)
  $8$
4. D)
  $10$
5. E)
  $13$
5.2013 AMC 10A #19
In base $10$ , the number $2013$ ends in the digit $3$ . In base $9$ , on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit $6$ . For how many positive integers $b$ does the base- $b$ -representation of $2013$ end in the digit $3$ ?
1. A)
  $6$
2. B)
  $9$
3. C)
  $13$
4. D)
  $16$
5. E)
  $18$
6.2013 AMC 10B #18
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 $1000$ share this property?
1. A)
  $33$
2. B)
  $34$
3. C)
  $45$
4. D)
  $46$
5. E)
  $58$
7.2017 AMC 10B #14
An integer $N$ is selected at random in the range $1\leq N \leq 2020$ . What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$ ?
1. A)
  $\frac{1}{5}$
2. B)
  $\frac{2}{5}$
3. C)
  $\frac{3}{5}$
4. D)
  $\frac{4}{5}$
5. E)
  $1$
8.2018 AMC 10B #13
How many of the first $2018$ numbers in the sequence $101, 1001, 10001, 100001, \dots$ are divisible by $101$ ?
1. A)
  $253$
2. B)
  $504$
3. C)
  $505$
4. D)
  $506$
5. E)
  $1009$
9.2020 AMC 10A #18
Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}.$ For how many such quadruples is it true that $a\cdot d-b\cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0\cdot 1-3\cdot 1 = -3$ is odd.)
1. A)
  $48$
2. B)
  $64$
3. C)
  $96$
4. D)
  $128$
5. E)
  $192$
10.2020 AMC 10B #12
The decimal representation of $\frac{1}{20^{20}}$ consists of a string of zeros after the decimal point, followed by a $9$ and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
1. A)
  $23$
2. B)
  $24$
3. C)
  $25$
4. D)
  $26$
5. E)
  $27$
11.2021 AMC 10B #16
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 $15$ ?
1. A)
  $4$
2. B)
  $5$
3. C)
  $6$
4. D)
  $7$
5. E)
  $8$
12.2023 AMC 10B #15
What is the least positive integer $m$ such that $m \cdot 2! \cdot 3!\cdot 4!\cdot 5! \dots 16!$ is a perfect square?
1. A)
  $30$
2. B)
  $30030$
3. C)
  $70$
4. D)
  $1430$
5. E)
  $1001$
13.2025 AMC 10B #8
Emmy says to Max, "I ordered $36$ math club sweatshirts today." Max asks, "How much did each shirt cost?" Emmy responds, "I'll give you a hint. The total cost was $\$ \underline A~\underline B~\underline B.\underline B~\underline A$ , where $A$ and $B$ are digits and $A \neq 0$ ." After a pause, Max says, "That was a good price." What is $A + B$ ?
1. A)
  $7$
2. B)
  $8$
3. C)
  $11$
4. D)
  $14$
5. E)
  $15$
14.2025 AMC 10A #17
Let $N$ be the unique positive integer such that dividing $273436$ by $N$ leaves a remainder of $16$ and dividing $272760$ by $N$ leaves a remainder of $15$ . What is the tens digit of $N$ ?
1. A)
  $0$
2. B)
  $1$
3. C)
  $2$
4. D)
  $3$
5. E)
  $4$
15.2015 AMC 10B #25
A rectangular box measures $a \times b \times c$ , where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$ . The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?
1. A)
  $4$
2. B)
  $10$
3. C)
  $12$
4. D)
  $21$
5. E)
  $26$
16.2017 AMC 10B #25
Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100$ , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$ . What was her score on the sixth test?
1. A)
  $92$
2. B)
  $94$
3. C)
  $96$
4. D)
  $98$
5. E)
  $100$
17.2018 AMC 10B #21
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 $323$ ?
1. A)
  $324$
2. B)
  $330$
3. C)
  $340$
4. D)
  $361$
5. E)
  $646$
18.2018 AMC 10B #23
How many ordered pairs $(a, b)$ of positive integers satisfy the equation $a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$ where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$ , and $\text{lcm}(a,b)$ denotes their least common multiple?
1. A)
  $\text{ 0}$
2. B)
  $\text{ 2}$
3. C)
  $\text{ 4}$
4. D)
  $\text{ 6}$
5. E)
  $\text{ 8}$
19.2018 AMC 10A #22
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 $a$ ?
1. A)
  $\text{ 5}$
2. B)
  $\text{ 7}$
3. C)
  $\text{ 11}$
4. D)
  $\text{ 13}$
5. E)
  $\text{ 17}$
20.2021 AMC 10A #22
Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$ , the second sheet contains pages $3$ and $4$ , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$ . How many sheets were borrowed?
1. A)
  $10$
2. B)
  $13$
3. C)
  $15$
4. D)
  $17$
5. E)
  $20$

Number Theory set — answersAMC 10 Step by Step

Name: Date: September 13, 2026

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Answers

#	Answer	Topic and the idea it turns on
1	D	Number Properties — n!(n+1)! = (n!)^2 (n+1), so the expression is a square exactly when (n+1)/2 is a square; 18/2 = 9 works.
2	B	Primes — The prime digits are 2, 3, 5, 7, so there are only 4 + 16 = 20 candidates; list them and test the sixteen two-digit ones for primality.
3	E	Fractions & Decimals — The error is the tail 0.00ababab..., which equals ab/9900; so 66 * ab/9900 = 0.5 gives ab = 75.
4	B	Primes — Using 2 makes the sum even, so take five odd primes; 3+5+7+11+13 = 39 is composite, 41 is unreachable, and 3+5+7+11+17 = 43 is prime.
5	C	Bases & Digits — The last base-b digit of 2013 is its remainder mod b, so b must divide 2010 = 235*67 (16 divisors) with b > 3.
6	D	Bases & Digits — For 1bcd the condition is d = 1 + b + c with b + c <= 8, giving 45 pairs; the only extra number in the 2000s is 2002.
7	D	Modular Arithmetic — Fourth powers of 1, 2, 3, 4 are all 1 mod 5, so N^16 is 1 mod 5 exactly when 5 does not divide N.
8	C	Modular Arithmetic — Terms are 10^k + 1 for k = 2 to 2019; since 100 is -1 mod 101, divisibility by 101 happens exactly when k is 2 mod 4.
9	C	Number Properties — ad - bc is odd exactly when one product is odd and the other even; a product is odd for 4 of 16 pairs, so 2(4)(12) = 96.
10	D	Fractions & Decimals — Write 20^20 = 2^20 * 10^20; since 2^20 is about 1.05 million, 1/2^20 is about 9.5 * 10^-7, then shift 20 more places.
11	C	Divisibility & Factors — Uphill numbers cannot end in 0, so the last digit is 5; choose a subset of {1,2,3,4} whose sum with 5 is a multiple of 3.
12	C	Number Properties — Pair (2k-1)! with (2k)! = (2k-1)! times 2k: the product is a square times 2 * 4 * ... * 16 = 2^8 * 8!, whose non-square part is 70.
13	C	Divisibility & Factors — The total in cents is ABBBA and must be divisible by 36 = 4 * 9; divisibility by 4 forces A even and by 9 forces 3 \| A.
14	E	GCD & LCM — N divides 273436 - 16 = 273420 and 272760 - 15 = 272745, hence their gcd 45; a remainder of 16 forces N > 16, leaving N = 45.
15	B	Diophantine Equations — Divide by abc to get 1/a + 1/b + 1/c = 1/2; ordering forces 3 <= a <= 6, then bound b likewise and solve for c.
16	E	Modular Arithmetic — Shift scores down by 90; the total is a multiple of 7 in [28, 49], and removing the seventh score 5 must leave a multiple of 6.
17	C	Divisibility & Factors — The next divisor d must share a factor with 323 = 17 * 19, since otherwise 323d > 9999 divides n; the smallest such d is 340.
18	B	GCD & LCM — Write a = gx, b = gy with x, y coprime; then 63 = g(xy(20 - g) + 12), so g divides 63 and only g = 21 works.
19	D	GCD & LCM — Exponents of 2 and 3 through the gcd chain force gcd(d,a) = 6k with k coprime to 6; between 70 and 100 only 78 = 6 * 13 fits.
20	B	Diophantine Equations — Equating remaining-page total to 19 times remaining-page count gives c(4b + 2c - 37) = 325; the only divisor c of 325 that fits within 25 sheets is 13.