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
How many ordered pairs of positive integers (x, y) satisfy 1/x + 1/y = 1/6?
Answer: 9
The equation becomes (x − 6)(y − 6) = 36, and 36 has 9 positive divisors.
What is lcm(12, 18, 30)?
Answer: 180
Take the highest prime powers: 2^2·3^2·5 = 180.
What is 2026 written in base 8?
Answer: 3752
3·512 + 7·64 + 5·8 + 2 = 1536 + 448 + 40 + 2 = 2026.
What is the remainder when 123456789 is divided by 11?
Answer: 5
The alternating digit sum from the right is 9−8+7−6+5−4+3−2+1 = 5.
Which statement is logically equivalent to "If P, then Q"?
Answer: Not P or Q.
P → Q is false only when P is true and Q is false, which matches ¬P ∨ Q.
How many perfect squares divide 2^4·3^6?
Answer: 12
Even exponents give 3 choices for 2 (0, 2, 4) and 4 choices for 3 (0, 2, 4, 6), so 3·4 = 12.
What is the remainder when 12! is divided by 13?
Answer: 12
By Wilson's theorem, (p − 1)! ≡ −1 (mod p), so 12! ≡ −1 ≡ 12 (mod 13).
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