PrepSwift # of Numbers in Factorials III quiz - Question 5

GRE GRE Quant Problem Solving
1

Question source: prepswift quiz for # of Numbers in Factorials III, question 5

The question: If the largest integer p such that 6^p divides 78!, and (2^q×3)^p divides 78!, are both the same, what is the largest possible integer q?

Multiple choice answers: 1, 2, 3, 4, and 5.

  1. There is a calculation error in the described solution: there are 74 multiples of 2 in 78! and not 73.
  2. The solution is unclear to me. Can anyone help me understand how to better cope with this question? Even reading the question itself is difficult, and I don’t know how to write it out and start solving.

For example, I eventually managed to solve it correctly but what I did first was calculate p and I got that it must be 36. Then you can check which q in the options allows p to still be the limiting factor, 74/q >= 36.

Why didn’t you solve for p first? Was it a mistake that I did?

2

Indeed, correct - this has been fixed.

And indeed, this is correct as well.

For example, I eventually managed to solve it correctly but what I did first was calculate p and I got that it must be 36.

I don’t understand what you mean by this - you can’t find p without assuming a value of q (or you’ll have a relationship in p and q).

3

Well, according to the first sentence, p is an integer such that 6^p divides 78! so you can find p and use it in the next part. Is that not right?

4

I see - the only problem with your approach is that it’s a bit less rigorous. For example, you could show that p = 36 only because 6 = 2 \times 3, and more importantly that the power of 3 is 1. The reason is that if it was, say , 18^p, this won’t work because the power of 3 is higher which you’ll need to take into account.

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