This remainder question confused me. Any shortcut to solve?

Question

For positive integers a and b, the remainder when a is divided by b is equal to the remainder when b is divided by a. Which of the following could be a value of ab ?
 I. 24
 II. 30
 III. 36
 A. II only
 B. III only
 C. I and II only
 D. II and III only
 E. I, II, and III

S Raghav · Accepted Answer

It sounds confusing, but here&rsquo;s a neat trick: the only way both remainders can be equal is when a and b are the same. If they&rsquo;re not equal, one remainder ends up being too large to make sense.
So just look for values of ab where a = b. That means ab must be a perfect square.
&bull; 24 &rarr; not a square &bull; 30 &rarr; not a square &bull; 36 &rarr; yes, 6 &times; 6
Only 36 works, so the answer is B. Shortcut: check which of the options is a perfect square &mdash; only that can be a &times; a.

B.K Sharma · Answer

Here&rsquo;s the quick insight: if the remainder when a is divided by b is equal to the remainder when b is divided by a, then the only way that works is if a = b.
If a &ne; b, say a < b, then remainder of a &divide; b is just a, which means b &divide; a would also have to give remainder a. But that can&rsquo;t happen &mdash; remainder can&rsquo;t be equal to or more than the divisor. So a = b is the only case that works.
That means ab must be a perfect square. Now just check which of the given options are perfect squares:

24 = not a square
30 = not a square
36 = 6 &times; 6 &rarr; valid

So the only possible value is III, and the answer is B.