Free AIME Number Theory Questions and Answers

Question 1

What is the greatest common divisor (GCD) of 252 and 198?

Accepted Answer

6

Answer

2

Answer

18

Answer

54

Question 2

If  𝑎 and 𝑏 are relatively prime, which of the following statements is true?

Accepted Answer

𝑎 and 𝑏 have no common divisors other than 1

Answer

𝑎 and 𝑏 are both prime numbers

Answer

𝑎 + 𝑏 is always even

Answer

𝑎 ⋅ 𝑏 is always a perfect square

Question 3

How many positive divisors does the number 360 have?

Accepted Answer

24

Answer

18

Answer

20

Answer

30

Question 4

What is the smallest positive integer 𝑥 such that 𝑥 ≡ 2 (mod 3) and 𝑥 ≡ 3 (mod5)?

Accepted Answer

8

Answer

28

Answer

13

Answer

23

Question 5

Find the least positive integer 𝑛 such that 𝑛 is congruent to 1 modulo 4, 2 modulo 5, and 3 modulo 6.

Accepted Answer

58

Answer

38

Answer

78

Answer

98