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

#Algebra #NumberTheory #ModularArithmetic #Help #Easymoney

Easy Math Editor

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Comments

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 - 6 years, 6 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

Trevor Arashiro - 6 years, 6 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.

Brian Charlesworth - 6 years, 6 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.

Calvin Lin Staff - 6 years, 6 months ago

@Calvin Lin – Unfortunately not.

Trevor Arashiro - 6 years, 6 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?

Calvin Lin Staff - 6 years, 6 months ago

Answers that we tried but weren't excepted

Trevor Arashiro - 6 years, 6 months ago

If it is "with remainder" is the answer infinity?

A Former Brilliant Member - 6 years, 6 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.

Brian Charlesworth - 6 years, 6 months ago

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

Gremlins?

Guiseppi Butel - 6 years, 6 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. :)

Brian Charlesworth - 6 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$