How many positive integers less than 103 have an odd number of positive divisors?

Only perfect squares have an odd number of factors. Factors always come in pairs except for perfect squares where there is an additional factor (the square root) which is on its own.

√1000 ≈ 31.62

Hence there will be 31 positive integers less than 1000 with an odd number of positive integer factors.

The only positive integers that have an odd number of positive divisors are perfect squares, so the only positive integers less than 103 that have an odd number of positive divisors are 1, 4, 9,. . ., 100. There are 10 of these numbers.

Step-by-step explanation:

Divde 13/143 with 13 = 1/11