Problem 6 · 2023 Math Kangaroo
Easy
Number Theory
divisibilityperfect-squareprimes
We call a positive integer n twoprime if it has exactly three different positive factors, namely 1, 2 and the number n itself. How many twoprime numbers are there?
Show answer
Answer: B — 1
Show hints
Hint 1 of 2
A number with exactly three divisors must be the square of a prime.
Still stuck? Show hint 2 →
Hint 2 of 2
Here the three divisors have to be exactly 1, 2 and n — so 2 must be the prime.
Show solution
Approach: a number with exactly three factors is a prime squared
- Exactly three positive factors means n = p2 for a prime p, with factors 1, p, p2.
- The factors must be 1, 2 and n, so p = 2 and n = 4 (factors 1, 2, 4).
- That is the only such number, so there is 1.
Mark:
· log in to save