49 = 7^2
Divisors: 1, 7, 49 -> exactly 3.
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GMAT Quant
Primes and Divisors
Recognize primes quickly and identify numbers with exactly three positive divisors.
Prime idea
A prime has exactly two positive divisors: 1 and itself.
Prime recognition supports factorization, divisor counts, and many number-property questions.
Primes below 30
- 1
- 2, 3, 5, 7
- 2
- 11, 13, 17, 19
- 3
- 23, 29
Primes from 30 to 60
- 1
- 31, 37
- 2
- 41, 43, 47
- 3
- 53, 59
Key rule
Exactly three positive divisors has a specific form.
A positive integer has exactly three positive divisors if and only if it equals p^2 for some prime p.
Why p^2 has three divisors.
- Number
- p^2
- Divisors
- 1, p, p^2
- Count
- 3
Square of a prime versus ordinary composite.
18
Has more than three divisors -> does not qualify.
Counting shortcut
To count such numbers up to N, count primes p with p^2 <= N.
You do not need to list every divisor of every candidate.
Mini application
Try this: how many positive integers <= 100 have exactly three positive divisors?
Count the prime squares.
Solution
There are 4: 4, 9, 25, and 49.
They are 2^2, 3^2, 5^2, and 7^2. The next prime square, 11^2, exceeds 100.
Remember this
Exactly three divisors means prime squared.
Convert the divisor-count question into a prime-recognition question.