COMC COMC Number Theory and Algebra

Question 1

What is the sum of all positive divisors of 28?

Accepted Answer

56

Answer

The divisors of 28 are 1, 2, 4, 7, 14, and 28, which sum to 56.

Question 2

What is the remainder when 7^100 is divided by 4?

Accepted Answer

1

Answer

Since 7 ≡ 3 (mod 4) and 3² ≡ 1 (mod 4), we get 7^100 ≡ (3²)^50 ≡ 1 (mod 4).

Question 3

How many positive divisors does 360 have?

Accepted Answer

24

Answer

Since 360 = 2³ × 3² × 5¹, the divisor count is (3+1)(2+1)(1+1) = 24.

Question 4

If gcd(a, b) = 6 and lcm(a, b) = 60, what is the product ab?

Accepted Answer

360

Answer

For any positive integers, gcd(a,b) × lcm(a,b) = ab, so ab = 6 × 60 = 360.

Question 5

What is the smallest positive integer that has exactly 5 positive divisors?

Accepted Answer

16

Answer

A number with exactly 5 divisors must be p^4 for a prime p; the smallest is 2^4 = 16.

COMC - Canadian Open Mathematics Challenge Practice Test