If m and n are two positive integers such that 5m + 7n = 46, then what

This is a special linear equation in 2 variables, with constraints on the variables. Because of these constraints, the equation can actually be solved.

Dividing the equation, 5m + 7n = 46 by 7, we have,

\(\frac{5m }{ 7}\) + n = \(\frac{42 }{ 7}\) +\( \frac{4 }{ 7}\)
Keeping n on the LHS, transferring \(\frac{5m }{ 7}\) to the RHS, we have,

n = 6 + \(\frac{4 }{ 7}\) – \(\frac{5m }{ 7 }\)

n = 6 + \(\frac{(4 – 5m) }{ 7}\)

n can only be an integer if (4 – 5m) is fully divisible by 7. The first value of m for which this happens is, m = 5.
When m = 5, n = 6 + \(\frac{(4 – 25) }{ 7}\) = 6 – 3 = 3.

The next value of m can be obtained by adding the coefficient of n i.e. 7 to the first value of m.
So, when m = 12, n = 6 + (4 – 60) 7 = 6 – 8 = -2.
It can be seen that n has reduced by 5 which is the coefficient of m.

The value by which one variable changes (increase / decrease) will be equal to the coefficient of the other variable. This is how special equations behave and it is this property which makes solving them a breeze, once the first set of values is found.

From our analysis of the equation, we see that we only have one set of positive integral values for m and n, i.e. m = 5 and n = 3.
The value of mn = 5 * 3 = 15.

The correct answer option is A.