Problem:
How many positive integers less than 50 have an odd number of positive integer divisors?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be a positive integer and a divisor of . Then is also : divisor of . Thus the divisors of occur in pairs and these two divisors will be distinct unless is a perfect square and . It follows that has an odd number of divisors if and only if is a perfect square. There are perfect squares among the numbers .
Note. If is an integer then where is the prime. The divisors of are those with for all . Thus, has divisors, a product which will be an odd number only when each is even.