Examine what is the smallest value of m if n^2 = 756m, m, n positve integers?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The smallest value of m has to be determined for which n^2 = 756*m. It is being assumed that n and m have to take on integral values; if that is not the case, there is no limit to how small a value m can equal.

756 = 189*4 = 9*4*21 = 3^2*2^2*7*3 = (3*2)^2*21

For n^2 = 756*m, with the condition that n and m are integers, the smallest value of m is 21.

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tonys538 | Student, Undergraduate | (Level 1) Valedictorian

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For two numbers m and n related by n^2 = 756m, we have to find the smallest value for m.

If n^2 = 756*m, when 756*m is expressed in the factorial form each integer should be present as a square.

The number 756 can be written as a product of integers as : 4*9*7*3 = 2^2*3^2*7*3

Now 4 and 9 are squares of integers. m should have a value such that there are two instances of the remaining factors 7 and 3 also.

If m = 21, 756*m = 4*9*49*9 = (2*3*7*3)^2 = n^2

The smallest value of m is 21.

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