AMC 8 Number Theory & Divisibility 3

Question 1

What is the prime factorization of 120?

Accepted Answer

2³ × 3 × 5

Answer

120 = 8 × 15 = 2³ × 3 × 5.

Question 2

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

Accepted Answer

14

Answer

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

Question 3

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

Accepted Answer

36

Answer

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

Question 4

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

Accepted Answer

6

Answer

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.

Question 5

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

Accepted Answer

84

Answer

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

AMC 8 - American Mathematics Competition 8 Practice Test