If a, b and c are positive integers, is a = b?

[VJesus12]

If a, b and c are positive integers, is a = b?

(1) (a + b)/(ab) is even.
(2) ac = c/b

The OA is the option D.

How can I show that both a and b are equal? Could someone give me some help, please? Thanks in advance.

[Vincen]

Hello Vjesus12.

Let's take a look at your question.

If a, b and c are positive integers, is a = b?

First statement

(1) (a + b)/(ab) is even.

Here we have the following: $$even=\frac{a+b}{ab}=\frac{a}{ab}+\frac{b}{ab}=\frac{1}{b}+\frac{1}{a}.$$ Since a and b are positive integers then 1/a and 1/b is less or equal than 1.

Also, their sum is less or equal than 1.

But, since 1/a + 1/b is an even number (an integer), then a=1 and b=1. Therefore a=b.

So, this statement is sufficient.

Second statement

(2) ac = c/b

Now we have $$ac=\frac{c}{b}\ \ \Rightarrow\ \ ab=1.$$ Since a and b are positive integers, then the only way to have ab=1 is when a=1 and b=1. Again, we get a=b.

So, this statement is also sufficient.

Therefore, the correct answer here is the option D.

I hope it helps.

[Jeff@TargetTestPrep]

If a, b and c are positive integers, is a = b?

(1) (a + b)/(ab) is even.
(2) ac = c/b

(a + b)/(ab) is even.

If a and b are positive integers, it's almost always the case that a + b < ab (i.e., (a + b)/(ab) is a proper fraction).

Thus, in order for (a + b)/(ab) to be an even integer, a and b must be very small. The smallest integer a and b can be is 1, and we see that (a + b)/(ab) = 2/1 = 2.

However, if one of them is 1 and the other is not, then (a + b)/(ab) is a fraction between 1 and 2.

If both a and b are 2, then (a + b)/(ab) = 4/4 = 1.

In other words, for any of the other pairs of positive integers a and b, we will have (a + b)/(ab) < 1. So the only way for (a + b)/(ab) to be even is if a = b = 1. We see that statement one alone is sufficient.

Statement Two Alone:

ac = c/b

That means ab = c/c or ab = 1. Since a and b are positive integers, the only way ab = 1 is a = b = 1. We see that statement two alone is also sufficient.

Answer: D