Question

The number of proper divisors of $35700$ which are divisible by $10$ and the sum of these divisors.

Answer 1

$35700=357\times 100$

$=3\times 119\times 5\times 5\times 2\times 2$

$=2^2\times 5^2\times 3\times {17}^2$

Sum of divisors $2^{(2+1)}{5}^{(2+1)}{3}^{(1+1)}{17}^{(2+1)}$

$=2^3\times 5^3\times 3^2\times {17}^3$

Now, sum of those excluding $(2\times 5)$ multiple

i.e. either not contain $2$ or $5$

$=5^3\times 3^2\times {17}^3+2^3\times 3^2\times {17}^3$

$\therefore$ Solution is $2^3\times 5^3\times 3^2\times {17}^3-[5^3\times 3^2\times {17}^3+2^3\times 3^2\times {17}^3]$

$=67980$

Hence, the answer is $67980.$