Question

Find the number of even proper divisors of 1008.
(a) 24
(b) 23
(c) 22
(d) none of these

Answer 1

Hint: First, define proper divisors. Then find the prime factorisation of 1008. Take a 2 common as we need even proper divisors. Then use the formula: to find the number of divisors from a prime factorization of the form ${{a}^{x}}{{b}^{y}}{{c}^{z}}$, we get the number of divisors = (x + 1)(y + 1)(z + 1). Subtract 1 from this to get the final answer.

Complete step-by-step answer:
In this question, we need to find the number of even proper divisors of the number 1008.
First, let us define proper divisors.
A positive proper divisor is a positive divisor of a number, excluding itself. 
So, the proper divisors of a number which are even are the even proper divisors of the number.
We will now find the prime factorisation of 1008.
The prime factorisation of 1008 is given by the following:
1008 = 2 $\times $ 2 $\times $ 2 $\times $ 2 $\times $ 3 $\times $ 3 $\times $ 7

Since, we only need even proper divisors, so we will take out one of the two common. So, after taking one of the twos common, we will have the following:
1008 = 2 (2 $\times $ 2 $\times $ 2 $\times $ 3 $\times $ 3 $\times $ 7)
This can be written as the following:
1008 = $2\times \left( {{2}^{3}}\times {{3}^{2}}\times 7 \right)$

By multiplying every divisor of the number within the brackets, we can find the even divisors of the given number as every number within the brackets will be multiplied by 2 which is taken out and gives the even divisors.

We know that, to find the number of divisors from a prime factorization of the form ${{a}^{x}}{{b}^{y}}{{c}^{z}}$, we get the number of divisors = (x + 1)(y + 1)(z + 1)
Using this formula for the expression in the bracket above, we get the following:
Number of even divisors of 1008 = (3 +1)(2 + 1)(1 +1) = 4 $\times $ 3 $\times $ 2 = 24
But the above divisors also contain the 1008 which is not a proper divisor.
So, the number of even proper divisors of 1008 is 23.

Hence, option (b) is correct.

Note: In this question, it is very important to know the following: a positive proper divisor is a positive divisor of a number , excluding itself, and to find the number of divisors from a prime factorization of the form ${{a}^{x}}{{b}^{y}}{{c}^{z}}$, we get the number of divisors = (x + 1)(y + 1)(z + 1).