Question

If $N=2^4\times 3^5\times 5^2\times 7^2$ then

• A. Number of even proper divisors of N are 215.
• B. Number of divisors of N divisible by 15 are 150.
• C. Number of ways in which the number N can be expressed as a product of two divisors are 135.
• D. Number of ways in which the number N can be expressed as a product of two divisors which are coprime are 8.

Answer 1

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

The correct options are
A Number of even proper divisors of N are 215.
B Number of divisors of N divisible by 15 are 150.
C Number of ways in which the number N can be expressed as a product of two divisors are 135.
D Number of ways in which the number N can be expressed as a product of two divisors which are coprime are 8.

Given, $N=2^4\times 3^5\times 5^2\times 7^2$
$\therefore$ Number of divisors$=(4+1)(5+1)(2+1)(2+1)=270$
$\therefore$ Number of ways in which the number $N$ can be expressed as a product of two divisors = $\frac{270}{2}$=135
$N=2(2^3\times 3^5\times 5^2\times 7^2)$
by multiplying every divisor of the number within the brackets, we can find the even divisors of the given number as every number with in the brackets will be multiplied by $2$ which is taken out and gives the even divisors
Number of even divisors $=(3+1)(5+1)(2+1)(2+1)=216$
this includes ${}^{\prime }{N}^{\prime }$ which is not a proper divisor
$\therefore$ Number of even proper divisors = 215
$N=15(2^4\times 3^4\times 5^1\times 7^2)$
$\therefore$ Number of divisors of N divisible by 15 = (4+1)(4+1)(1+1)(2+1)
=150
To find the Number of ways in which the number N can be expressed as a product of two divisors which are coprime, all the powers of a particular prime should be on the same divisor out of the two which are written as a product
So, the four primes with their powers should be arranged into two sets
There are $16$ ways of writing these subsets ($2^4$)
But, the divisor which is appeared in the subset is multiplied by the counter part which is in the same subset to give the number
So, Number of ways in which the number N can be expressed as a product of two divisors which are coprime are $8$.
$\therefore$ options A,B,C,D