how many divisor of 10010000 which are multiple of $52$?

Question

$$10010000=11\times7\times13\times2^4\times5^4$$

The Total number of divisors are $200$ then how to find which divisors are multiple of $52$? The divisors which are multiples of $52$ must be multiple of $2,4,13$

Answer 1

The notation $a\mid b$ means "$a$ divides $b$."

Hint: there is a bijection between $\{x$ such that $d\mid x\mid n\}$ and $\{$divisors of $n/d\}$.

(Explicitly, it is $x\mapsto x/d$. It has an obvious inverse map, hence bijectivity.)

Answer 2

So the question really is, how many divisors of $10010000/52 = 11\cdot 7 \cdot 2^2 \cdot 5^4$.

The divisors of a prime power is one more than the exponent, and between primes it is multiplicative, so the answer is $2\cdot 2\cdot 3 \cdot 5 = 60$

Answer 3

$52 = 2^2*13$

So diviser the are multiples of $52$ are the divisors in the form $k*2^2*13$ so $k $ may be composed of any of the remaining prime divisors: $11,7,2^2,5^4$. So there are $2*2*3*5=60$ possible values of $k $ so there are $60$ possible $k*52$ that divide $10010000$.