CAT 2017 Slot 1 QA Question 4: How many distinct positive integer-valued solutions exist to the equation $(x^2 - 7x + 11)^{x^2 - 13x + 42} = 1$?

Question

How many distinct positive integer-valued solutions exist to the equation $(x^2 - 7x + 11)^{x^2 - 13x + 42} = 1$?

Accepted Answer

For a power to equal 1, the possible cases here are: exponent 0, base 1, or base $-1$ with an even exponent.

Case 1: Exponent = 0
$x^2 - 13x + 42 = 0$
$\Rightarrow (x - 6)(x - 7) = 0$
$\Rightarrow x = 6, 7$

Case 2: Base = 1
$x^2 - 7x + 11 = 1$
$\Rightarrow x^2 - 7x + 10 = 0$
$\Rightarrow (x - 2)(x

How many distinct positive integer-valued solutions exist to the equation (x2−7x+11)x2−13x+42=1(x^2 - 7x + 11)^{x^2 - 13x + 42} = 1(x2−7x+11)x2−13x+42=1?