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If 180mn=k^3, where m, n and k are positive integers, what is the leas

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If 180mn=k^3, where m, n and k are positive integers, what is the leas  [#permalink]

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New post 23 Sep 2015, 02:26
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If 180mn=k^3, where m, n and k are positive integers, what is the least possible value of m+n?

A. 11
B. 20
C. 22
D. 25
E. 33

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Re: If 180mn=k^3, where m, n and k are positive integers, what is the leas  [#permalink]

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New post 23 Sep 2015, 02:28
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Senior Manager
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Re: If 180mn=k^3, where m, n and k are positive integers, what is the leas  [#permalink]

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New post 23 Sep 2015, 02:36
thanks, a friend of mine asked me to solve this for him. there was not any indication to where the q belongs to
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Re: If 180mn=k^3, where m, n and k are positive integers, what is the leas  [#permalink]

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New post 29 Sep 2015, 20:30
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Hi shasadou,

This question (and the original question that it's based on) is based on prime-factorization.

We're told that 180(M)(N) = K^3 and that M, N and K are all positive integers. To figure out the least possible value of M and N, we first have figure out the product of M and N.

180 can be prime factored into (2)(2)(3)(3)(5)

So (2)(2)(3)(3)(5)(M)(N) = K^3 = (K)(K)(K)

Since K represents the same integer in each pair of parentheses, each of those Ks must have the exact same prime-factorization. Since we're dealing with 2s, 3s and 5s....the smallest possible K MUST be (2)(3)(5).

So we need (2)(2)(2)(3)(3)(3)(5)(5)(5).....The M and the N have to contain the 'missing' 2, 3 and two 5s that we need.

Thus, (M)(N) = (2)(3)(5)(5) = 150

Now that we know the product, we can go about figuring the smallest possible value of M and N. Since they're integers, they could be...

1 and 150 (or the reverse)
2 and 75 (or the reverse)
3 and 50 ( " )
5 and 30 ( " )
6 and 25 ( " )
10 and 15 ( " )

You'll notice that the 'closer' the two numbers get together, the smaller their sum becomes. 10 and 15 is the closest that those two variables can get, so the smallest possible sum is 25.

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Re: If 180mn=k^3, where m, n and k are positive integers, what is the leas  [#permalink]

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Re: If 180mn=k^3, where m, n and k are positive integers, what is the leas   [#permalink] 24 Feb 2018, 08:54
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