Question

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

Answer 1

Given that,

$N=24500$

Part (1)

$24500=2\times 2\times 5\times 5\times 5\times 7\times 7$

$=2^2\times 5^3\times 7^2$

Number of ways

$=\frac{(powerof2+1)(powerof5+1)(powerof3+1)}{2}$

$=\frac{(2+1)(3+1)(2+1)}{2}$

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

$=18ways$

Part (2).

The number of ways by $5$N $=\frac{(powerof2+1)(powerof5+1)(powerof3+1)}{2}$$+5$

$=\frac{3\times 4\times 3}{2}+5$

$=18+5=23ways$

Hence, this is the answer