How many integers in the set 1, 2, 3......, 100 have exactly 3 divisors?

Question

How many integers in the set 1, 2, 3......, 100 have exactly 3 divisors?

1. A. 4 (Correct answer)
2. B. 8
3. C. 6
4. D. 2

Correct Answer

Option A — 4

Detailed Solution & Explanation

The correct answer is 4.

Key Points

• A number has exactly three divisors only when it is the square of a prime. If n = p², its divisors are 1, p and p² — precisely three.
• Any other form gives a different count: a prime has 2 divisors, p³ has 4, and pq has 4.
• Squares of primes up to 100:
  • 2² = 4, 3² = 9, 5² = 25, 7² = 49
• The next is 11² = 121, which exceeds 100 — so there are 4 such integers.

Additional Information

• The divisor-count formula: for n = p₁^a × p₂^b × …, the number of divisors is (a+1)(b+1)…. Three divisors requires (a+1) = 3, so a = 2 and a single prime factor.
• Because 3 is prime, the count cannot factor any other way — which is why only perfect squares of primes qualify.
• By the same reasoning, exactly two divisors means a prime, and exactly four means either p³ or a product of two distinct primes.
• Related: the number of divisors is odd if and only if n is a perfect square, since divisors otherwise pair off.