Question

Let $N=2^6{.5}^5{.7}^6{.10}^7$, then the total number of even factors of $N$ is

• A. $1183$
• B. $1200$
• C. $1050$
• D. $540$

Answer 1

The correct option is A $1183$

The prime factorization of $2^6{.5}^5{.7}^6{.10}^7$ is $2^{13}{.5}^{12}{.7}^6$.
The total number of factors of $N=(13+1)(12+1)(6+1)$
$=\mathrm{14.13.7}$
We need to find the total of even factors. For this, let us find the total number of odd factors and then subtract this from the total number of factors. Any odd factor will have to be a combination of powers of only $5$ and $7$.
Total number of odd factors of $2^{13}{.5}^{12}{.7}^6=(12+1)(6+1)=13\times 7$
Total number of factors $=(13+1)(12+1)(6+1)$
Total number of even factors $=14\times 13\times 7-13\times 7$
Number of even factors $=13\times 13\times 7=1183$