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Re: If 5400mn = k4, where m, n, and k are positive integers [#permalink]
13 Feb 2011, 13:43

3

This post received KUDOS

Expert's post

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banksy wrote:

If 5400mn = k4, where m, n, and k are positive integers, what is the least possible value of m + n? (A) 11 (B) 18 (C) 20 (D) 25 (E) 33

banksy please format the questions properly!

Question should read: If 5400mn = k^4, where m, n, and k are positive integers, what is the least possible value of m + n? A. 11 B. 18 C. 20 D. 25 E. 33

Note that m, n, and k are positive integers.

First of all: \(5,400=2^3*3^3*5^2\). Now, in order \(5,400mn=2^3*3^3*5^2*m*n\) to be equal to the integer in fourth power then \(mn\) must complete the powers of 2, 3 and 5 to the fourth power, hence the least value of \(mn\) for which \(2^3*3^3*5^2*m*n=k^4\) is for \(mn=2*3*5^2=150\). In this case \(5,400mn=2^3*3^3*5^2*(2*3*5^2)=(2*3*5)^4=k^4\).

So we have that the least value of \(mn\) is \(2*3*5^2\). Next: in order to minimize \(m+n\) we should break \(2*3*5^2\) into two multiples which are closest to each other: \(2*5=10\) and \(3*5=15\), their sum is \(10+15=25\).

Re: If 5400mn = k4, where m, n, and k are positive integers [#permalink]
27 Feb 2011, 02:38

Bunuel wrote:

banksy wrote:

If 5400mn = k4, where m, n, and k are positive integers, what is the least possible value of m + n? (A) 11 (B) 18 (C) 20 (D) 25 (E) 33

banksy please format the questions properly!

Question should read: If 5400mn = k^4, where m, n, and k are positive integers, what is the least possible value of m + n? A. 11 B. 18 C. 20 D. 25 E. 33

Note that m, n, and k are positive integers.

First of all: \(5,400=2^3*3^3*5^2\). Now, in order \(5,400mn=2^3*3^3*5^2*m*n\) to be equal to the integer in fourth power then \(mn\) must complete the powers of 2, 3 and 5 to the fourth power, hence the least value of \(mn\) for which \(2^3*3^3*5^2*m*n=k^4\) is for \(mn=2*3*5^2=150\). In this case \(5,400mn=2^3*3^3*5^2*(2*3*5^2)=(2*3*5)^4=k^4\).

So we have that the least value of \(mn\) is \(2*3*5^2\). Next: in order to minimize \(m+n\) we should break \(2*3*5^2\) into two multiples which are closest to each other: \(2*5=10\) and \(3*5=15\), their sum is \(10+15=25\).

Answer: D.

I did not understand the following parts. m*n=2*3*5^2=150, and So we have that the least value of \(mn\) is \(2*3*5^2\). Next: in order to minimize \(m+n\) we should break \(2*3*5^2\) into two multiples which are closest to each other: \(2*5=10\) and \(3*5=15\), their sum is \(10+15=25\). _________________

Re: If 5400mn = k4, where m, n, and k are positive integers [#permalink]
27 Feb 2011, 02:56

Expert's post

Baten80 wrote:

Bunuel wrote:

banksy wrote:

If 5400mn = k4, where m, n, and k are positive integers, what is the least possible value of m + n? (A) 11 (B) 18 (C) 20 (D) 25 (E) 33

banksy please format the questions properly!

Question should read: If 5400mn = k^4, where m, n, and k are positive integers, what is the least possible value of m + n? A. 11 B. 18 C. 20 D. 25 E. 33

Note that m, n, and k are positive integers.

First of all: \(5,400=2^3*3^3*5^2\). Now, in order \(5,400mn=2^3*3^3*5^2*m*n\) to be equal to the integer in fourth power then \(mn\) must complete the powers of 2, 3 and 5 to the fourth power, hence the least value of \(mn\) for which \(2^3*3^3*5^2*m*n=k^4\) is for \(mn=2*3*5^2=150\). In this case \(5,400mn=2^3*3^3*5^2*(2*3*5^2)=(2*3*5)^4=k^4\).

So we have that the least value of \(mn\) is \(2*3*5^2\). Next: in order to minimize \(m+n\) we should break \(2*3*5^2\) into two multiples which are closest to each other: \(2*5=10\) and \(3*5=15\), their sum is \(10+15=25\).

Answer: D.

I did not understand the following parts. m*n=2*3*5^2=150, and So we have that the least value of \(mn\) is \(2*3*5^2\). Next: in order to minimize \(m+n\) we should break \(2*3*5^2\) into two multiples which are closest to each other: \(2*5=10\) and \(3*5=15\), their sum is \(10+15=25\).

As explained before in order \(5,400mn=2^3*3^3*5^2*m*n\) to be equal to the integer in fourth power then \(mn\) must complete the powers of 2, 3 and 5 to the fourth power er (well generally to the multiple of 4, though as we need the least value of mn then to 4), hence the least value of \(mn\) for which \(2^3*3^3*5^2*m*n=k^4\) is for \(mn=2*3*5^2=150\). In this case \(5,400mn=2^3*3^3*5^2*(2*3*5^2)=(2*3*5)^4=k^4\): mn must have one 2 to complete 2^3 to 2^4, one 3 to complete 3^3 to 3^4 and two 5's to complete 5^2 to 5^4.

So we have that the least value of \(mn\) is \(2*3*5^2\). Next: in order to minimize \(m+n\) (taking into account that \(mn=2*3*5^2\)) we should break \(2*3*5^2\) into two multiples which are closest to each other: \(2*5=10\) and \(3*5=15\), their sum is \(10+15=25\) (all other break downs of \(mn=2*3*5^2=150\) will have the greater sum: 1+150=151, 2+75=77, 3+50=52, ...).

Re: If 5400mn = k^4, where m, n, and k are positive integers [#permalink]
18 Jun 2014, 14:31

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