Question

# In how many ways can the number 7056 be resolved into two factors?

Hint: In this question we need to find the number of ways in which the given number can be resolved into two factors. So, firstly we would be doing prime factorization of the given number and then use the formula for the same to resolve it into two factors. This would help us find the answer.

$7056 = {2^4} \times {3^2} \times {7^2}$
So, the given number is of form ${a^p}{b^q}{c^r}.....$where a, b, c…. are prime numbers and p, q, r…. are all even numbers.
So, we can resolve the number into two factors in $\dfrac{1}{2}\left[ {\left( {p + 1} \right)\left( {q + 1} \right)\left( {r + 1} \right)....... + 1} \right]$ ways.
So, the number of ways in which the given number can be resolved into two factors is $\dfrac{1}{2}\left[ {\left( {4 + 1} \right)\left( {2 + 1} \right)\left( {2 + 1} \right) + 1} \right]$
$= \dfrac{1}{2}\left[ {5 \times 3 \times 3 + 1} \right]$
$= \dfrac{{46}}{2}$
$= 23$