What is the greatest value of integer n such that is a factor of ?
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Wait, I got 11 but I saw people saying 12... let me recheck. I counted powers of 2 in 26! as 13+6+3+1 = 23, then divided by 2 to get 11. Where does 12 come from?
Yeah 4^n = 2^(2n), so 2n ≤ 23 gives n ≤ 11.5, so n = 11. Not sure how anyone gets 12.
I think some people just divide 23 by 2 and round up by mistake. Floor it.
Nice one, this is really a disguised 'count the twos' problem. Took me a sec to realize 4^n means I need pairs of 2s.
Right, that's the whole trick. Once you see 4 = 2^2 it's straightforward.
Is this really 655-705 level? Seems more like a standard trailing-zeros style counting problem to me.
I think the trap of forgetting to divide by 2 is what bumps it up a bit.
Thanks, clear explanation. I initially forgot to use Legendre's formula and just tried multiplying 4s until I passed 26!, which was a mess lol.
What it tests
Your command of divisibility, primes, factors, parity, and how the GMAT tests properties of integers.
Common trap
Forgetting that 1 is not prime, that 0 is divisible by every integer, or that negative numbers flip your assumptions.