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Number Theory & Divisibility Flashcards

6 cards from real AMC 8 practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 6 Number Theory & Divisibility flashcards as text

  1. What is the prime factorization of 120?

    Answer: 2³ × 3 × 5

    120 = 8 × 15 = 2³ × 3 × 5.

  2. How many perfect squares are there from 1 to 200, inclusive?

    Answer: 14

    The perfect squares from 1² = 1 to 14² = 196 are all at most 200, giving 14 perfect squares.

  3. What is the GCF of 2³ × 3² and 2² × 3⁴?

    Answer: 36

    The GCF takes the minimum exponent for each prime: 2² × 3² = 4 × 9 = 36.

  4. What is the smallest positive integer n such that n² is divisible by 12?

    Answer: 6

    Since 12 = 2² × 3, for 12 | n², n must be divisible by both 2 and 3; the smallest such n is 6, and 6² = 36 = 12 × 3.

  5. How many three-digit numbers have digits that are strictly increasing from left to right (each digit greater than the one before)?

    Answer: 84

    Each set of 3 distinct digits from 1–9 arranged in increasing order forms exactly one such number, giving C(9,3) = 84.

  6. What is the remainder when 2²⁵ is divided by 3?

    Answer: 2

    Powers of 2 mod 3 alternate: 2¹≡2, 2²≡1, 2³≡2, …; odd powers give remainder 2, so 2²⁵ ≡ 2 (mod 3).