Problem 1 · 1996 AJHSME
Easy
Number Theory
divisor-counting
How many positive factors of 36 are also multiples of 4?
Show answer
Answer: B — 3.
Show hints
Hint 1 of 2
A number that is both a factor of 36 AND a multiple of 4 must already have the 4 built in. So write it as 4 × (something) — what does that 'something' have to be?
Still stuck? Show hint 2 →
Hint 2 of 2
Pull the 4 out front: the number is 4 × (a factor of 36 ÷ 4 = 9). Counting them becomes just counting the factors of 9 — a much smaller job.
Show solution
Approach: factor out the required 4
- A multiple of 4 looks like 4 × k. For it to also divide 36, the leftover k must divide 36 ÷ 4 = 9. So instead of hunting through all of 36's factors, we just need the factors of 9.
- 9 has factors 1, 3, 9 — three of them. Multiplying each by 4 gives 4, 12, 36, so there are 3 such numbers.
- Why this transfers: 'count the multiples of d that also divide N' always becomes 'count the factors of N ÷ d.' Pulling out the forced factor shrinks the problem every time.
Mark:
· log in to save