How many divisors of 10^5 will have at least one zero at its end?

Solution:

In order for a divisor (or any number) to have a zero at its end, it must have a 10 as a factor, i.e., a 2-and-5 pair. Notice that

10^5 = 2^5 x 5^5 = (2^4 x 5^4) x (2 x 5)

Therefore, any divisors of 2^4 x 5^4 will have a zero at its end when multiplied by 2 x 5. Since 2^4 x 5^4 has (4 + 1)(4 + 1) = 25 divisors, 10^5 has 25 divisors that have at least one zero at its end.

Alternate Solution:

Notice that 10^5 = 2^5 x 5^5 has (5 + 1)(5 + 1) = 36 divisors in total.

A divisor of 10^5 does not have a zero at its end if it is a divisor of 2^5 or if it is a divisor of 5^5. Notice that both 2^5 and 5^5 have 5 + 1 = 6 divisors. Since 1 is the only common divisor of 2^5 and 5^5, the set of numbers which are either a divisor of 2^5 or a divisor of 5^5 has 6 + 6 - 1 = 11 elements.

Subtracting the number of divisors that do not have a zero at its end from the total number of divisors, we see that 36 - 11 = 25 divisors of 10^5 will have a zero at its end.

Answer: D

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