Question

For which of the following values of m is m to power (91) + 2m to power (90) the square of an integer?

a) 20
b) 24
c) 27
d) 30
e) 34
explain

Answer 1

We can rewrite m^91 + 2m^90 as (m+2m^90)^2 - m^90.

We are looking for values of m such that m+2m^90 is an integer.

For m = 30, m+2m^90 = 30 + 230^90 = 30 + 2(30^90) = 60, which is an integer.

For m = 24, m+2m^90 = 24 + 224^90 = 24 + 2(24^90) = 48, which is also an integer.

For the other values of m, m+2m^90 is not an integer. Therefore, the only values of m for which m^91 + 2m^90 is the square of an integer are m = 24 and m = 30.