The source: PrepSwift # of numbers in factorials III - question 6
The question is: If the largest integer p such that 7^p divides n! is x and 7^p divides (n+1)! is (x+3), what is the smallest possible value of n?
I understand the part of the solution that says that for this to be true, n+1 must be a multiple of 7^3, but then I checked the answer by plugging in n=342 and got two completely different p’s. According to the instructions isn’t p supposed to stay the same in both calculations? Am I misunderstanding the instructions?
I don’t understand - what are you getting? Are you looking at a different question?
if n=342 then the largest possible p for which 7^p divides n! is 54.
but the largest possible p for which 7^p divides (n+1)! is 57
My problem might be a comprehension problem (I’m not a native English speaker). My understanding of the first part of the question is that for n=342 the same p is supposed to give x for 342/(&^p) and x+3 for 343/(7^p).
The sentence itself in the question is not complete. It starts with “if the largest p…” and then two conditions, but no conclusion for the “if”. Is it supposed to be “The largest integer p is such that… two conditions”, and then the question? And if it is then p should always be the same number, right?
Anyways, this is all very confusing…
So the intention was to read it like this:
- the largest integer p such that 7^p divides n! is x
- the largest integer p such that 7^p divides (n + 1)! is x + 3
In other words, the p can differ in the two statement. So if x = 54, that means p = 54 for statement 1, and x = 57 in statement 2 which means that p = 57 for that statement.
Is that not how you’re seeing it? To be clear, I’m not ruling out that the question might have a readability issue.
Yeah, I didn’t read it like that. I find it strange that a variable will have 2 different solutions in the same question. I understood it like it’s a given that there is 1 p and both these statements are true.
Anyway I understand now, thanks!