Sonia2023 wrote:
banksy wrote:
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
chetan2u Bunuel - for minimum values of two numbers whose product is fixed , we generally need the numbers to be as close as possible.
So, if the number is a perfect square, we can easily take out minimum values. But for the numbers which are not perfect squares, can we not take square root of the product and then solve it as follows:
\(\sqrt{150}\) = 5 \(\sqrt{6}\)
5\(\sqrt{ 6}\)+ 5\(\sqrt{ 6}\) = minimum value of the sum
2*5* \(\sqrt{6}\)
10*\(\sqrt{6}\)
25
Did this method worked only for this question or can we use this one for other questions as well?
Thank you
The method would require knowing square roots and can become calculation intensive.
If the options had 24 too, what would you have answered. A quicker way using your approach would be.
The number is \(5\sqrt{6}\)
Now, we know \(\sqrt{6}\) should be between \(\sqrt{4}\) or 2 and \(\sqrt{9}\) or 3.
Thus, look for factors to the in the range or closest to range 5*2 to 5*3.
150 has factors 10 and 15 close to that range.
However best is to look for prime factors as shown above
OR
Or 150 lies close to 144, which is square of 12. So, the closest factors should lie on either side of the 12.
Which factor is closest to 12 => you can make out that it is 10, and from 10, you can find the other factor 10*15
_________________