Question

The difference of number of even and odd factors of 10478160 is

• A. $9^3$
• B. $4^4$
• C. $8^3$
• D. $6^3$

Answer 1

The correct option is D $6^3$

If any number x can be expressed in its prime factorisation form as $x={\left(p\right)}^{a_1}{\left(q\right)}^{a_2}{\left(r\right)}^{a_3}$
where, p is even and q, r are odd numbers, then, the total number of factors of the given number x are $(a_1+1)(a_2+1)(a_3+1).$

Also, the number of odd factors of the given number x will be $(a_2+1)(a_3+1).$

Here, $x=10478160=2^4\times 3^5\times 5^1\times 7^2\times {11}^1$

Total number of factors of 10478160 = (4 + 1)(5 + 1)(1 + 1)(2 + 1)(1 + 1)

= 5 × 6 × 2 × 3 × 2

= 360

Number of odd factors of 10478160 = (5 + 1)(1 + 1)(2 + 1)(1 + 1)

= 6 × 2 × 3 × 2

= 72

∴ Number of even factors of 10478160

= Total factors of 10478160 – Number of odd factors of 10478160

= 360 – 72

= 288

∴ Difference of number of even and odd factors of 10478160

= Number of even factors of 10478160 – Number of odd factors of 10478160

= 288 – 72

= 216

= $2^3$ × $3^3$

= $6^3$

Hence, the correct answer is option (d).