Question

Write the set of the odd divisors of 72.

Answer 1

Hint: List all the factors of the number 72 by finding all those numbers from 1-72 with which 72 is divisible. Alternatively, remove all the powers of 2 from 72 and find the divisors of the number. Argue why the divisors of the new number will all be odd and why the set of the divisors so formed will include all the odd divisors of 72.

Complete step-by-step answer:

A divisor of a number is a number which when divides the number leaves no remainder,i.e. a is a divisor of b if and only if there exists an integer k such that b = ak. Mathematically we write a|b which is read as “a divides b.”
72 is divisible by 1,2,3,4,,6,8,9,12,18,24,36 and 72.
Among these, the odd divisors are 3,9.
Hence the set of odd divisors of 72 is {1,3,9}.

Note: Alternatively we have
$72={{2}^{3}}{{3}^{2}}$
Hence the divisors of ${{3}^{2}}$ will be the odd divisors of 72. This is because we have removed all the powers of two and hence any divisor will be odd. Also, odd divisors do not have any power of 2. Hence the set of odd divisors of 72 and the set of divisors of ${{3}^{2}}$ will be the same.
Now the set of divisors of ${{3}^{2}}$ is {1,3,9}
Hence the set of odd divisors of 72 is {1,3,9}