For each positive integer k, let . Which of the following is the greatest value of n such that divides ?
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Wait, so a_k = 7k for k = 1 to 28? Then the product is just 7^28 times 28!, right? The 7s don't matter for factors of 10, so I just need the number of trailing zeros in 28!. Is that the whole trick here?
That's the cleanest way to see it. Took me a second to notice the 7s are irrelevant.
Got 5 the first time by just counting multiples of 5 and forgetting to add the extra from 25. Classic trap. Feels like this one is right in that 600-650 range where they punish shortcuts.
Yeah, that extra factor from 25 is the whole point of the question I think.
Quick method: floor(28/5) + floor(28/25) = 5 + 1. But why do we only count the 5s and not the 2s? Because there are way more 2s than 5s in 28!, so 5s are the limiting factor. Thanks for posting!
Exactly, always the 5s that limit the number of 10s.
Is the answer 6? I keep getting 5 but one of my friends said 6 and now I'm second guessing. Can someone confirm without spoiling it fully?
Recount the multiples of 25 carefully, that's usually where it flips.
What it tests
Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.
Common trap
Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.