🔢 Number-Theory Warmups
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Basic Divisibility · Easy · source: Original (AMC-style)
Is 15 divisible by 3?
A) Yes B) No C) Sometimes D) Cannot be determined E) Depends on context
Answer & Solution
Answer: A
Yes, because $15 = 3 \times 5$.
2.
Tags: Prime Numbers · Easy · source: Original (AMC-style)
Which of the following is prime?
A) $4$ B) $6$ C) $7$ D) $8$ E) $9$
Answer & Solution
Answer: C
7 is prime because its only divisors are 1 and 7.
3.
Tags: Even Numbers · Easy · source: Original (AMC-style)
What is the smallest even number greater than 10?
A) $10$ B) $11$ C) $12$ D) $13$ E) $14$
Answer & Solution
Answer: C
12 is the smallest even number greater than 10.
4.
Tags: Odd Numbers · Easy · source: Original (AMC-style)
What is the largest odd number less than 20?
A) $17$ B) $18$ C) $19$ D) $20$ E) $21$
Answer & Solution
Answer: C
19 is the largest odd number less than 20.
5.
Tags: GCD · Easy · source: Original (AMC-style)
What is the GCD of 12 and 18?
A) $2$ B) $3$ C) $6$ D) $9$ E) $12$
Answer & Solution
Answer: C
The divisors of 12 are 1,2,3,4,6,12 and of 18 are 1,2,3,6,9,18. The largest common divisor is 6.
6.
Tags: LCM · Easy · source: Original (AMC-style)
What is the LCM of 4 and 6?
A) $12$ B) $18$ C) $24$ D) $30$ E) $36$
Answer & Solution
Answer: A
The multiples of 4 are 4,8,12,16,... and of 6 are 6,12,18,24,.... The smallest common multiple is 12.
7.
Tags: Modular Arithmetic · Easy · source: Original (AMC-style)
What is $17 \bmod 5$?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: B
$17 = 3 \times 5 + 2$, so $17 \bmod 5 = 2$.
8.
Tags: Remainders · Easy · source: Original (AMC-style)
What is the remainder when 23 is divided by 7?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: B
$23 = 3 \times 7 + 2$, so the remainder is 2.
9.
Tags: Factors · Easy · source: Original (AMC-style)
How many factors does 12 have?
A) $3$ B) $4$ C) $5$ D) $6$ E) $7$
Answer & Solution
Answer: D
The factors of 12 are 1,2,3,4,6,12, so there are 6 factors.
10.
Tags: Perfect Squares · Easy · source: Original (AMC-style)
What is $5^2$?
A) $10$ B) $15$ C) $20$ D) $25$ E) $30$
Answer & Solution
Answer: D
$5^2 = 5 \times 5 = 25$.
11.
Tags: Perfect Cubes · Easy · source: Original (AMC-style)
What is $3^3$?
A) $9$ B) $12$ C) $18$ D) $27$ E) $36$
Answer & Solution
Answer: D
$3^3 = 3 \times 3 \times 3 = 27$.
12.
Tags: Number Bases · Easy · source: Original (AMC-style)
What is 101 in binary equal to in decimal?
A) $3$ B) $4$ C) $5$ D) $6$ E) $7$
Answer & Solution
Answer: C
$101_2 = 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = 5$.
13.
Tags: Divisibility Rules · Easy · source: Original (AMC-style)
Is 123 divisible by 3?
A) Yes B) No C) Sometimes D) Cannot be determined E) Depends on context
Answer & Solution
Answer: A
Yes, because the sum of digits is $1+2+3=6$, which is divisible by 3.
14.
Tags: Prime Factorization · Easy · source: Original (AMC-style)
What is the prime factorization of 20?
A) $2 \times 10$ B) $4 \times 5$ C) $2^2 \times 5$ D) $2 \times 2 \times 5$ E) Both C and D
Answer & Solution
Answer: E
Both $2^2 \times 5$ and $2 \times 2 \times 5$ are correct forms of the prime factorization.
15.
Tags: Basic Number Theory · Easy · source: Original (AMC-style)
What is the sum of the first 5 positive integers?
A) $10$ B) $12$ C) $15$ D) $18$ E) $20$
Answer & Solution
Answer: C
The sum is $1+2+3+4+5 = 15$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | A | C | C | C | C | A | B | B | D | D | D | C | A | E | C |