COMC COMC Number Theory and Algebra 1 ā Questions and Answers
Question 1: What is the sum of all positive divisors of 28?
- 56 (Correct answer)
- 28
- 42
- 48
Correct answer: 56
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?
- 1 (Correct answer)
- 3
- 0
- 2
Correct answer: 1
Since 7 ā” 3 (mod 4) and 3Ā² ā” 1 (mod 4), we get 7^100 ā” (3Ā²)^50 ā” 1 (mod 4).
Question 3: How many positive divisors does 360 have?
- 24 (Correct answer)
- 18
- 12
- 36
Correct answer: 24
Since 360 = 2Ā³ Ć 3Ā² Ć 5Ā¹, 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?
- 360 (Correct answer)
- 300
- 420
- 180
Correct answer: 360
For any positive integers, gcd(a,b) Ć lcm(a,b) = ab, so ab = 6 Ć 60 = 360.
Question 5: What is the smallest positive integer that has exactly 5 positive divisors?
- 16 (Correct answer)
- 12
- 20
- 24
Correct answer: 16
A number with exactly 5 divisors must be p^4 for a prime p; the smallest is 2^4 = 16.
Question 6: What is the remainder when 2026Ā² is divided by 10?
- 6 (Correct answer)
- 4
- 2
- 8
Correct answer: 6
The last digit of 2026 is 6, and 6 raised to any positive power always ends in 6, so the remainder is 6.
COMC COMC Number Theory and Algebra 1
What is the sum of all positive divisors of 28?