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

## Comments

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TopNewestDoesn'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, 3 months ago

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But that's what I was thinking as well. Unfortunately, it's supposedly not the answer – Trevor Arashiro · 2 years, 3 months ago

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– Calvin Lin Staff · 2 years, 3 months ago

Do you know what the answer is supposed to be? That could offer a way to backtrack and figure out what they mean.Log in to reply

– Trevor Arashiro · 2 years, 3 months ago

Unfortunately not.Log in to reply

– Brian Charlesworth · 2 years, 3 months ago

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.Log in to reply

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

Gremlins? – Guiseppi Butel · 2 years, 3 months ago

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– Brian Charlesworth · 2 years, 3 months ago

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. :)Log in to reply

If it is "with remainder" is the answer infinity? – Anup Navin · 2 years, 3 months ago

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– Brian Charlesworth · 2 years, 3 months ago

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.Log in to reply

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, 3 months ago

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– Trevor Arashiro · 2 years, 3 months ago

Answers that we tried but weren't exceptedLog in to reply