Question

Which of the following have odd number of factors?

• A. 64
• B. 81
• C. 50
• D. 52

Answer 1

Answer: (A), (B)

The correct options are
A

64

B

81

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 are 7, which is odd because 8 times 8 gives us 64 and hence the factor 8 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.
Now, in the options given, 64 and 81 are squares of the numbers 8 and 9 respectively, and hence have odd number of factors.