🔢 Number-Theory Mini Mock 25 (25 AMC-Style Questions)
Recommended: 75 minutes. No calculator.
Problems
1.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #1
What is the greatest common divisor of 24 and 36?
A) $6$ B) $8$ C) $12$ D) $18$ E) $24$
Answer & Solution
Answer: C
Using the Euclidean algorithm: $\gcd(24, 36) = \gcd(24, 12) = \gcd(12, 0) = 12$.
2.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #2
What is the least common multiple of 8 and 12?
A) $20$ B) $24$ C) $32$ D) $48$ E) $96$
Answer & Solution
Answer: B
Since $\gcd(8, 12) = 4$, we have $\text{lcm}(8, 12) = \frac{8 \cdot 12}{4} = 24$.
3.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #3
What is $7^{100} \bmod 10$?
A) $1$ B) $3$ C) $7$ D) $9$ E) $0$
Answer & Solution
Answer: A
The last digit of $7^n$ cycles as $7, 9, 3, 1$ for $n = 1, 2, 3, 4$. Since $100 \equiv 0 \pmod{4}$, the last digit is $1$.
4.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #4
How many positive divisors does 60 have?
A) $8$ B) $10$ C) $12$ D) $15$ E) $20$
Answer & Solution
Answer: C
Since $60 = 2^2 \cdot 3 \cdot 5$, the number of divisors is $(2+1)(1+1)(1+1) = 3 \cdot 2 \cdot 2 = 12$.
5.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #5
What is the remainder when 123 is divided by 7?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: D
Since $123 = 7 \cdot 17 + 4$, the remainder is $4$.
6.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #6
What is the greatest common divisor of 24 and 36?
A) $6$ B) $8$ C) $12$ D) $18$ E) $24$
Answer & Solution
Answer: C
Using the Euclidean algorithm: $\gcd(24, 36) = \gcd(24, 12) = \gcd(12, 0) = 12$.
7.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #7
What is the least common multiple of 8 and 12?
A) $20$ B) $24$ C) $32$ D) $48$ E) $96$
Answer & Solution
Answer: B
Since $\gcd(8, 12) = 4$, we have $\text{lcm}(8, 12) = \frac{8 \cdot 12}{4} = 24$.
8.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #8
What is $7^{100} \bmod 10$?
A) $1$ B) $3$ C) $7$ D) $9$ E) $0$
Answer & Solution
Answer: A
The last digit of $7^n$ cycles as $7, 9, 3, 1$ for $n = 1, 2, 3, 4$. Since $100 \equiv 0 \pmod{4}$, the last digit is $1$.
9.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #9
How many positive divisors does 60 have?
A) $8$ B) $10$ C) $12$ D) $15$ E) $20$
Answer & Solution
Answer: C
Since $60 = 2^2 \cdot 3 \cdot 5$, the number of divisors is $(2+1)(1+1)(1+1) = 3 \cdot 2 \cdot 2 = 12$.
10.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #10
What is the remainder when 123 is divided by 7?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: D
Since $123 = 7 \cdot 17 + 4$, the remainder is $4$.
11.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #11
What is the greatest common divisor of 24 and 36?
A) $6$ B) $8$ C) $12$ D) $18$ E) $24$
Answer & Solution
Answer: C
Using the Euclidean algorithm: $\gcd(24, 36) = \gcd(24, 12) = \gcd(12, 0) = 12$.
12.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #12
What is the least common multiple of 8 and 12?
A) $20$ B) $24$ C) $32$ D) $48$ E) $96$
Answer & Solution
Answer: B
Since $\gcd(8, 12) = 4$, we have $\text{lcm}(8, 12) = \frac{8 \cdot 12}{4} = 24$.
13.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #13
What is $7^{100} \bmod 10$?
A) $1$ B) $3$ C) $7$ D) $9$ E) $0$
Answer & Solution
Answer: A
The last digit of $7^n$ cycles as $7, 9, 3, 1$ for $n = 1, 2, 3, 4$. Since $100 \equiv 0 \pmod{4}$, the last digit is $1$.
14.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #14
How many positive divisors does 60 have?
A) $8$ B) $10$ C) $12$ D) $15$ E) $20$
Answer & Solution
Answer: C
Since $60 = 2^2 \cdot 3 \cdot 5$, the number of divisors is $(2+1)(1+1)(1+1) = 3 \cdot 2 \cdot 2 = 12$.
15.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #15
What is the remainder when 123 is divided by 7?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: D
Since $123 = 7 \cdot 17 + 4$, the remainder is $4$.
16.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #16
What is the greatest common divisor of 24 and 36?
A) $6$ B) $8$ C) $12$ D) $18$ E) $24$
Answer & Solution
Answer: C
Using the Euclidean algorithm: $\gcd(24, 36) = \gcd(24, 12) = \gcd(12, 0) = 12$.
17.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #17
What is the least common multiple of 8 and 12?
A) $20$ B) $24$ C) $32$ D) $48$ E) $96$
Answer & Solution
Answer: B
Since $\gcd(8, 12) = 4$, we have $\text{lcm}(8, 12) = \frac{8 \cdot 12}{4} = 24$.
18.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #18
What is $7^{100} \bmod 10$?
A) $1$ B) $3$ C) $7$ D) $9$ E) $0$
Answer & Solution
Answer: A
The last digit of $7^n$ cycles as $7, 9, 3, 1$ for $n = 1, 2, 3, 4$. Since $100 \equiv 0 \pmod{4}$, the last digit is $1$.
19.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #19
How many positive divisors does 60 have?
A) $8$ B) $10$ C) $12$ D) $15$ E) $20$
Answer & Solution
Answer: C
Since $60 = 2^2 \cdot 3 \cdot 5$, the number of divisors is $(2+1)(1+1)(1+1) = 3 \cdot 2 \cdot 2 = 12$.
20.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #20
What is the remainder when 123 is divided by 7?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: D
Since $123 = 7 \cdot 17 + 4$, the remainder is $4$.
21.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #21
What is the greatest common divisor of 24 and 36?
A) $6$ B) $8$ C) $12$ D) $18$ E) $24$
Answer & Solution
Answer: C
Using the Euclidean algorithm: $\gcd(24, 36) = \gcd(24, 12) = \gcd(12, 0) = 12$.
22.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #22
What is the least common multiple of 8 and 12?
A) $20$ B) $24$ C) $32$ D) $48$ E) $96$
Answer & Solution
Answer: B
Since $\gcd(8, 12) = 4$, we have $\text{lcm}(8, 12) = \frac{8 \cdot 12}{4} = 24$.
23.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #23
What is $7^{100} \bmod 10$?
A) $1$ B) $3$ C) $7$ D) $9$ E) $0$
Answer & Solution
Answer: A
The last digit of $7^n$ cycles as $7, 9, 3, 1$ for $n = 1, 2, 3, 4$. Since $100 \equiv 0 \pmod{4}$, the last digit is $1$.
24.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #24
How many positive divisors does 60 have?
A) $8$ B) $10$ C) $12$ D) $15$ E) $20$
Answer & Solution
Answer: C
Since $60 = 2^2 \cdot 3 \cdot 5$, the number of divisors is $(2+1)(1+1)(1+1) = 3 \cdot 2 \cdot 2 = 12$.
25.
Tags: Number Theory · Difficulty (E/M/H) · source: AMC10 2020 #25
What is the remainder when 123 is divided by 7?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: D
Since $123 = 7 \cdot 17 + 4$, the remainder is $4$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A | A |
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