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Is there an answer?

Recently, on one of my math competitions, there was a problem that no one got, in clouding the teacher. So I was wondering if it's even answerable.

A number n has 4 distinct prime factors. Find the number of positive integers that divide n with remainder.

The answer is not:

N-16.

194

Note by Trevor Arashiro
2 years, 1 month ago

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Doesn't this depend on the multiplicity of the distinct prime factors?

Are we looking at \(n = p_{1}^{a_{1}} * p_{2}^{a_{2}} * p_{3}^{a_{3}} * p_{4}^{a_{4}}\) for distinct primes \(p_{k}\) with exponents \(a_{k} \ge 1\)?

If so, then the I think the desired number is \(\prod_{k=1}^4 (a_{k} + 1)\). Brian Charlesworth · 2 years, 1 month ago

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@Brian Charlesworth Sorry, but I just realized I worded the question slightly wrong. Its supposed to be "with remainder". So for this case, its just \(n-\prod_{k=1}^4(a_k+1)\)

But that's what I was thinking as well. Unfortunately, it's supposedly not the answer Trevor Arashiro · 2 years, 1 month ago

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@Trevor Arashiro Do you know what the answer is supposed to be? That could offer a way to backtrack and figure out what they mean. Calvin Lin Staff · 2 years, 1 month ago

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@Calvin Lin Unfortunately not. Trevor Arashiro · 2 years, 1 month ago

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@Trevor Arashiro Weird. Is there any chance that the source of the supposed answer is incorrect? I see now how the "n - 16" and "194" answers arose; you are looking at \(2*3*5*7 = 210\). This number definitely has \(16\) divisors, and thus there would be \(210 - 16 = 194\) positive integers less than \(210\) that would divide into \(210\) with non-zero remainder. Brian Charlesworth · 2 years, 1 month ago

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I wasn't aware that I re-shared this. If I did, I certainly didn't intend to.

Gremlins? Guiseppi Butel · 2 years, 1 month ago

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@Guiseppi Butel Haha Actually, there have been some gremlins lurking on Brilliant the past few days. Strange things have been happening when posting/editing solutions, causing format changes, symbols appearing out of nowhere, etc.. So yes, your unintended re-share could have been the work of gremlins. :) Brian Charlesworth · 2 years, 1 month ago

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If it is "with remainder" is the answer infinity? Anup Navin · 2 years, 1 month ago

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@Anup Navin That's a thought, but the 'standard' interpretation of "positive integer \(m\) divides \(n\) with remainder" assumes that \(m \le n\). Since it is stated that \(n\) has \(4\) distinct prime factors I think that we can assume that this standard interpretation is being applied, otherwise this prime information would have been largely irrelevant. Brian Charlesworth · 2 years, 1 month ago

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What kind of answer is expected? Is it multiple choice, integer answer, short algebraic answer, or long proof form?

Also, what does N-16 and 194 refer to? Calvin Lin Staff · 2 years, 1 month ago

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@Calvin Lin Answers that we tried but weren't excepted Trevor Arashiro · 2 years, 1 month ago

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