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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.

N-16.

194

Note by Trevor Arashiro
2 years, 7 months 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 $$\prod_{k=1}^4 (a_{k} + 1)$$. · 2 years, 7 months ago

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 · 2 years, 7 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. Staff · 2 years, 7 months ago

Unfortunately not. · 2 years, 7 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. · 2 years, 7 months ago

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

Gremlins? · 2 years, 7 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. :) · 2 years, 7 months ago

If it is "with remainder" is the answer infinity? · 2 years, 7 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. · 2 years, 7 months ago

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? Staff · 2 years, 7 months ago