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

Let's look at an example, say 18. The factors of 18 are 1, 2, 3, 6, 9 and 18. Notice that they come in pairs as
$18=1\times 18$
$18=2\times 9$
$18=3\times 6$, and so the number of factors of 18 is even.
Consider the factors of 64
$64=1\times 64$
$64=2\times 32$
$64=4\times 16$
$64=8\times 8$, and the number of factors is 7, which is odd because 8 times 8 gives us 64 and it is considered only once .
So if $m$ is a factor of $n$, then there exists an integer k such that $n=km$. This adds two numbers to the list of factors of $n$ unless $k=m$. Thus the number of factors is even unless $n=k\times k$, that is $n$ is a square.