DS-Number Properties : Need simple explanation to solve

This topic has expert replies
Junior | Next Rank: 30 Posts
Posts: 26
Joined: 04 Dec 2008
Location: Bangalore
Thanked: 1 times
If x is a positive integer, is x square - 1 divisible by 24?

(1) x is odd

(2) x leaves a remainder when divided by 3


Question: Not able to understand on how to solve and need explanation for the same.
Experts please help.


User avatar
GMAT Instructor
Posts: 2583
Joined: 02 Jun 2008
Location: Toronto
Thanked: 1090 times
Followed by:355 members
GMAT Score:780

by Ian Stewart » Tue Aug 20, 2019 5:38 am
A number will be divisible by 24 if it is divisible by 3 and by 8. We want to know if x^2 - 1 = (x-1)(x+1) is divisible by 24. Notice that x-1 and x+1 are separated by 2, so they are either consecutive odd integers or consecutive even integers.

From Statement 1, we learn that x is odd, so x-1 and x+1 must be consecutive even integers. If you look at any two consecutive even integers, one of them will be divisible by 2, and the other will be divisible by 4 (at least), so the product of two consecutive even integers is always divisible by 8. So Statement 1 ensures that (x-1)(x+1) is divisible by 8, but we don't know if it's divisible by 3, and Statement 1 is not sufficient.

From Statement 2, we learn that x is not divisible by 3. But x-1, x and x+1 are three consecutive integers, and among any three consecutive integers, you always find exactly one multiple of 3, since multiples of 3 are three apart. If x is not a multiple of 3, then either x-1 or x+1 must be. So Statement 2 ensures that (x-1)(x+1) is divisible by 3, but we don't know if it's divisible by 8, and Statement 2 is not sufficient.

Using both Statements, we know (x-1)(x+1) = x^2 - 1 is divisible by 8 from Statement 1, and by 3 from Statement 2, and is therefore divisible by 24, so the answer is C.
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com