Question

In how many ways can 2744 be resolved as a product of two factors?
A. 16
B. 14
C. 6
D. 8

Answer 1

Hint: First of all, find the prime factorization of 2744 by finding its prime factors. Then find the number of factors of 2744. And the required answer is obtained by dividing the number of factors of 2744 with 2 as they should be resolved into two products.

Complete step by step solution:
Consider the prime factors of 2744 i.e., \[2744 = 2 \times 2 \times 2 \times 7 \times 7 \times 7\]
So, the prime factorisation of \[2744 = {2^3} \times {7^3}\]
We know that if the prime factorisation of a number \[N = {a^m} \times {b^n}\] then the number of factors is given by \[\left( {m + 1} \right) \times \left( {n + 1} \right)\].
Therefore, the number of factors of 2744 are \[\left( {3 + 1} \right) \times \left( {3 + 1} \right) = 4 \times 4 = 16\].
And the number of ways it can be resolved by a product of two factors \[ = \dfrac{{16}}{2} = 8.\]
Thus, the correct option is D. 8

Note: If the prime factorisation of a number \[N = {a^m} \times {b^n}...............\] then the factors is given by \[\left( {m + 1} \right) \times \left( {n + 1} \right)...................\] and the number of ways it can be resolved by a product of \[p\] factors are given by \[\dfrac{{\left( {m + 1} \right) \times \left( {n + 1} \right).............................}}{p}\].