Question

Let N=24500, then find
i) The number of ways by which N can be resolved into two factors.
ii) The number of ways by which 5N can be resolved into two factors.

Answer 1

We have

$N=24500$

$=100\times 245=100\times 5\times 49$

$={\left(5\right)}^2\times {\left(2\right)}^2\times {\left(7\right)}^2$

$={\left(2\right)}^2\times {\left(5\right)}^2\times {\left(7\right)}^2$

Now

(i) No. of ways by which $N$ cam be resolved into two factors $=\frac{1}{2}\left[\left(2+1\right)\left(3+1\right)\left(2+1\right)\right]$

$=\frac{1}{2}\times 3\times 4\times 3=18$

again

$5N={\left(2\right)}^2\times {\left(5\right)}^4\times {\left(7\right)}^2$

No. of ways by which $5N$ by which can be resolved into two factors $=\frac{1}{2}\left[\left(2+1\right)\left(4+1\right)\left(2+1\right)+1\right]$

$=\frac{1}{2}\left[3\times 5\times 3+1\right]$

$=\frac{46}{2}=23$

Hence, the find the required answer.