Question

There are two integer values of n that
yield an integer value for $\frac{12-\sqrt{n} }{4-\sqrt{7} }$.
What is the sum of these two values of n?

Answer 1

Answer : 207

$4-\sqrt{7}$ divides $12-\sqrt{n}$

$12-\sqrt{n}$ equals 0

In the first case, as 12 is the multiple of 4

$12-\sqrt{n} =3(4-\sqrt{7} )$

solve for n

$12-\sqrt{n} =12-3\sqrt{7}$

$n=3^2*7$

∴$n=63$

Second case, if we solve for n

$12-\sqrt{n} =0$

$\sqrt{n} =12$

∴$n=144$

The answer is $207$.