QR question

[asckwan]

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Both 5^2 and 3^3 are factors of n*2^5*6^2*7^3. What is the smallest possible positive value of n? 25, 27, 45, 75, 125

 

[Chga]

woah kinda hard for a DAT problem w/o calculator 😀

 

[asckwan]

You're right about the numbers you used, but in the problem it's 3^3, 27.

I tried using a calculator to calculate it but none of them worked.

First, 5^2=25 3^3=27

next, n*2^5*6^2*7^3 = n*32*36*343 = 395136n

So if n=25, the equation would yield 9878400 which can't be it since 27 won't divide into that evenly.

If n=27, the equation would yield 1066932 which can't be it since 25 won't divide into that evenly.

And so on and so forth.

I think it might be a typo, but without the answer choices there still must be a correct numerical answer. Can anyone tell me a short cut way to figure it out?

 

[luder98]

Both 5^2 and 3^3 are factors of n*2^5*6^2*7^3. What is the smallest possible positive value of n? 25, 27, 45, 75, 125

I think this is how you do it...

Since 5^2 and 3^3 are factors of n*2^5*6^2*7^3, there must be a positive integer m that satisfies the below equation:

m*5^2*3^3 = n*2^5*6^2*7^3 (1)

Notice that 6^2 = (3*2)^2 = 3^2*2^2

Equation (1) becomes:

m*5^2*3^3 = n*2^5*3^2*2^2*7^3 or
m*5^2*3 = n*2^7*7^3 or
m = (n*2^7*7^3)/(5^2*3)

Since m is an integer, (n*2^7*7^3)/(5^2*3) must be an integer. For this to happen, 5^2*3 must be a factor of n. Thus, the smallest n is that factor, which is 25*3 = 75.

 

[asckwan]

Thanks so much luder! Never would have thought to do it that way. I appreciate your help!