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# [Check] AMC 8 Grinding Practice Question

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How many factors of 900 have exactly 18 factors

WHAT I DID (If I got it incorrect tell me what I got wrong)

Prime factorize 900..?

3^2 * 2^2 * 5^2

You find the number of factors of 900 by taking the exponents, adding one to each, and multiplying them out.

3 * 3 * 3 = 27. 900 has 27 factors..

We can apply the same rule to find the numbers that have 18 factors by working backward.

We prime factor 18.

3 * 3 * 2

So it must be in the form:

x^2 * y^2 * z

Values of x, y, z, must be subbed in with 3, 2, 5.

We find the number of ways that can be arranged. So its, just 3!

So 6?

OMG I am so scared right now I want to do good on the AMC 8. I have like anxiety issues or something because MY BRAIN IS SCREWED up and my HEART is EXPLODING

Nov 15, 2019
edited by CalculatorUser  Nov 15, 2019

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I know what I did wrong

Oh ok, I know what I did wrong.

Basically I got to a certain point where 2, 3, 5, can be substituted in here:

\(x^2*y^2*z\)
Then, I find all the possible values, and count them.

I thought it was just 3!, but if you calculated it out, it actually is 3.

Nov 15, 2019