Problem Solving

If \(a^2b\) is an integer which of the following must be an integer?

A. \(a\)
B. \(b\)
C. \(ab\)
D. \(b^2\)
E. None of the above

Answer: E

Source: Magoosh

In this one, we need to approach the PS with the elimination method. In that, we need to prove that a \(a^2b\) can be an integer without the given option being an integer.

But even before that, you can easily eliminate B and D, with help or an exponent property. That only case where \(n^2\) be is an integer when 'n' is an integer.

So if b is an integer, \(b^2\) has to be an integer and vice versa.
The question can only have 1 answer, so two solutions can be eliminated together as impossibles.

Now testing options 'A' and 'C'
Option A
a must be an integer
let's suppose a=0.5
so \(a^2\) = 0.25
If b is 4;
\(a^2b\) = 0.25 x 4 = 1 (an integer)
so eliminate

Option C
ab must be an integer
let's suppose a=0.5
and suppose b=4
So ab = 0.5 x 4 = 2 (an integer)
and using the math done above; \(a^2b\) = 0.25 x 4 = 1 (an integer)
so eliminate

so A, B, C, D being eliminated, the correct answer is E