Question

Choose the set of all numbers between 1 and 70 which have odd number of factors?

• A. 5, 10, 16, 30, 36, 55, 78, 95
• B. 1, 4, 9, 25, 36, 16, 49, 64
• C. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 57, 61, 67
• D. 1, 4, 9, 25, 36, 16, 49, 64, 81, 100

Answer 1

The correct option is B 1, 4, 9, 25, 36, 16, 49, 64

Factors come in pairs and for every number other than a perfect square, these pairs have different factor values in them.
For example: 18, which is not a perfect square, has the factor pairs as (1, 18), (2, 9), (3, 6) with different factor values, thus indicating that the number of factors in 18 is even.
For perfect squares, however, one of the factor pairs has a repeating factor value and thus has only 1 unique factor in that pair.
For example: 64, which is a perfect square, has the factor pairs as (1, 64), (2, 32), (4, 16), (8, 8). Here, the factor (8, 8) has a repeating factor value 8 and hence this makes the total number of factors for a perfect square to be odd.
Based on this, the set of all numbers between 1 and 70 which have odd number of factors is the set of perfect squares between 1 and 70.
So, perfect squares between 1 and 70 are 1, 4, 9, 16, 25, 36, 49, 64.