Data Sufficiency - Highest Power of 5 Dividing n!

Question

What is the highest power of 5 that can completely divide $n!$ ?

Statement I: Precisely 2 integers between 1 and $n$ , both inclusively are multiples of 50. Statement II: $n < 200$ .

Choose: (a) if the question can be answered by one statement alone, but cannot be answered by using the other statement alone. (b) if the question can be answered by using either statement alone. (c) if the question can be answered by using both the statements together but cannot be answered by using either statement alone. (d) if the question cannot be answered even by using both the statements together.

Accepted Answer

D. Statement I: Precisely 2 multiples of 50 means $100 \le n < 150$. For $n=100$, highest power of 5 dividing $100!$ is $\lfloor 100/5 \rfloor + \lfloor 100/25 \rfloor = 20 + 4 = 24$. But for $n=105$, exponent of 5 is $\lfloor 105/5 \rfloor + \lfloor 105/25 \rfloor = 21 + 4 = 25$. Thus $n$ is not uniquely determined. Statement II: $n < 200$ gives a vast range. Combining both: $100 \le n < 150$, which still gives different exponents of 5 depending on $n$. Hence, cannot be answered even using both statements.

Answer

if the question can be answered by one statement alone, but cannot be answered by using the other statement alone.

Answer

if the question can be answered by using either statement alone.

Answer

if the question can be answered by using both the statements together but cannot be answered by using either statement alone.

Answer

if the question cannot be answered even by using both the statements together.

Data Sufficiency - Highest Power of 5 Dividing n!

Explanation

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