Question

The number of ways in which 2160 can be written as product of two positive integers is:

• A. 40
• B. 20
• C. 80
• D. 100

Answer 1

Answer: (B)

20

Number of ways of expressing a given number as a product of two factors

$N=ap.bq.cr$ where $a,b,c$ are prime factors of $N$ and $p,q,r$ are positive integers) can be expressed as the product of two factors in different ways. The number of ways in which this can be done is given by the expression $=1/2\left\{\right(p+1\left)\right(q+1\left)\right(r+1\left)\right\}$ If $p,q,r$ etc. are all even, then the product $(p+1)(q+1)(r+1)$... becomes odd and the above rule will not be valid.

If $p,q,r$... are all even, it means that $N$ is a perfect square.

So, to find out the number of ways in which a perfect square can be expressed as a product of $2$ factors, we have the following $2$ rules

(1) As a product of two DIFFERENT factors is $\left(1/2\right)\left\{\right(p+1\left)\right(q+1\left)\right(r+1)...-1\}$ ways (excluding $X$)

(2) As a product of two factors (including $X$) is $\left(1/2\right)\left\{\right(p+1\left)\right(q+(r+1).+1\}$ way

When $n=2160=2^4\times 3^3\times 5d=(4+1)(3+1)(1+1)=40$

We have 40 divisors. So the Number of ways in which this can be written as a product of two numbers $=\frac{(4+1)(3+1)(1+1)}{2}=\frac{40}{2}=20$