Find the number of divisors

Number Theory Level pending

What is the least number of positive divisors of $x$ , if $x \gt 1$ , and $x^{2/3} , x^{5/12}$ and $x^{9/14}$ are all integers?

The answer is 85.

1 solution

Maria Kozlowska
Feb 12, 2017

$LCM(3,12,14)=84$

$x=p^{84}$ where $p$ is prime number will result in the following:

$x^{2/3}=p^{56}$

$x^{5/12}=p^{35}$

$x^{9/14}=p^{54}$

$p^{84}$ has $84+1=\boxed{85}$ divisors.

More info can be found here

If that's the intended solution, the problem should read x^(2/3) instead of x^2/3, and same with each of the others. Per order of operations, exponents happen before division, meaning parentheses are necessary (or appropriate formatting as in your solution) in order to communicate a fraction in the exponent.

Steve McMath - 4 years, 3 months ago

You can check here: rational exponents or any other place for the notation. If it is superscripted it is ok. If is not, it can be misunderstood. You can create a report on that.

Maria Kozlowska - 4 years, 3 months ago

Yes, I agree with your use of the notation. It's the original problem that didn't superscript properly. I hoped the writer would see the comment and correct it on his own. Not actually sure how to create a report.

Steve McMath - 4 years, 3 months ago

@Steve McMath – Click three dots at the bottom right corner of the problem window and choose "view reports".

Maria Kozlowska - 4 years, 3 months ago

@Maria Kozlowska – Ah, thanks. Looks like someone already did, but now I know.

Steve McMath - 4 years, 3 months ago

@Steve McMath – Thanks. I see that this problem has been edited.

As Maria has already pointed out, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the menu. This will notify the problem creator (and eventually staff) who can fix the issues.

Brilliant Mathematics Staff - 4 years, 3 months ago

Find the number of divisors

The answer is 85.

1 solution

0 pending reports