Number Theory & Mathematical Logic Flashcards
7 cards from real AMC 12 practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.
Read the first 7 Number Theory & Mathematical Logic flashcards as text
What is the remainder when 2^100 is divided by 7?
Answer: 2
Since 2^3 ≡ 1 (mod 7) and 100 = 3·33 + 1, 2^100 ≡ 2^1 = 2 (mod 7).
How many positive divisors does 360 have?
Answer: 24
360 = 2^3·3^2·5, so it has (3+1)(2+1)(1+1) = 24 divisors.
What is gcd(2^12 − 1, 2^18 − 1)?
Answer: 63
gcd(2^a − 1, 2^b − 1) = 2^gcd(a,b) − 1 = 2^6 − 1 = 63.
What are the last two digits of 7^2026?
Answer: 49
7^4 = 2401 ≡ 1 (mod 100) and 2026 ≡ 2 (mod 4), so 7^2026 ≡ 7^2 = 49 (mod 100).
What is the contrapositive of "If n^2 is even, then n is even"?
Answer: If n is odd, then n^2 is odd.
The contrapositive of P → Q is ¬Q → ¬P, giving "if n is not even, then n^2 is not even."
How many trailing zeros does 100! have?
Answer: 24
By Legendre's formula, ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24.
What is the smallest positive integer x with x ≡ 2 (mod 3), x ≡ 3 (mod 5), and x ≡ 2 (mod 7)?
Answer: 23
x ≡ 2 (mod 21) gives 2, 23, 44, …, and 23 ≡ 3 (mod 5).
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