Question

Let $P=780$, then the correct option(s) among the following is/are

• A. Sum of even divisors of $P$ is $1008$
• B. Sum of odd divisors of $P$ is $336$
• C. Total number of ways in which $P$ can be represented as product of $2$ coprime numbers is $8$
• D. Total number of ways in which $P$ can be represented as product of $2$ natural numbers is $12$

Answer 1

Answer: (B), (C), (D)

Total number of ways in which $P$ can be represented as product of $2$ natural numbers is $12$

$780=2^2\cdot 3\cdot 5\cdot 13$
For the divisors to be even, we should take atleast one $2$.
$\therefore$ Sum of even divisors
$=(2^1+2^2)(3^0+3^1)(5^0+5^1)({13}^0+{13}^1)=6\times 4\times 6\times 14=2016$

For the divisors to be odd, we should not take any $2$.
$\therefore$ Sum of odd divisors
$=(3^0+3^1)(5^0+5^1)({13}^0+{13}^1)=4\times 6\times 14=336$

Here, the number of different prime factors are $4$
$\therefore$ number of ways in which $780$ can be resolved as product of two coprime numbers is $2^{4-1}=8$

Since, $P$ is not a perfect square, the number of ways in which $P$ can be resolved as product of two numbers $=\frac{1}{2}\left[\right(2+1\left)\right(1+1\left)\right(1+1\left)\right(1+1\left)\right]=12$