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 Euler's totient φ(100)?
Answer: 40
φ(100) = 100·(1 − 1/2)(1 − 1/5) = 40.
What is the sum of all positive divisors of 72?
Answer: 195
72 = 2^3·3^2, so σ(72) = (1+2+4+8)(1+3+9) = 15·13 = 195.
How many positive integers n ≤ 100 are divisible by 3 or 5?
Answer: 47
By inclusion-exclusion, 33 + 20 − 6 = 47.
Which is the negation of "For every integer n, n^2 ≥ n"?
Answer: There exists an integer n such that n^2 < n.
Negating a universal statement gives an existential statement with the inner claim negated.
What is the units digit of 3^2026?
Answer: 9
Units digits of powers of 3 cycle 3, 9, 7, 1, and 2026 ≡ 2 (mod 4), so the digit is 9.
What is the multiplicative inverse of 7 modulo 26?
Answer: 15
7·15 = 105 = 4·26 + 1, so 7·15 ≡ 1 (mod 26).
What is the largest integer k such that 2^k divides 50!?
Answer: 47
⌊50/2⌋+⌊50/4⌋+⌊50/8⌋+⌊50/16⌋+⌊50/32⌋ = 25+12+6+3+1 = 47.
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