Question

Find the number of divisors and the sum of divisors of number $1400$.

Answer 1

Step 1: Calculate the factors

The given number is $1400$, and it can be expressed as,

$1400=2\times 2\times 2\times 5\times 5\times 7$

$=2^3\times 5^2\times 7^1$

Step 2: Calculate the number of divisors

Let the number $n=p^a{q}^b{r}^c$, ($p,q,r$ are prime numbers and $a,b,c$ are the powers)

Then the total number of divisors of a number $n$ where $n$ can be represented as powers of prime numbers is given by $(a+1)(b+1)(c+1)$.

Here $a=3,b=2,c=1$

The number of divisors $=(3+1)(2+1)(1+1)=24$

Step 3: Calculate the sum of divisors

Sum of divisors $=(p^0+p^1+p^2+..p^a)(q^0+q^1+q^2+..q^b)(r^0+r^1+r^2+..r^c)$

$=(2^0+2^1+2^2+2^3)(5^0+5^1+5^2)(7^0+7^1)$

$=15\times 31\times 8$

$=3720$

Hence, the number of divisors are $24$ and sum of divisors is $3720$.