Number Properties – Mathematical Operations on Terms with Remainders

Our Divisibility and Remainders modules discusses all the concepts of this topic in detail. We have seen how to handle remainders with mathematical operations on terms. Let’s take a look at an application of that today.

Say “” gives you a remainder of 2 when divided by 6. What will be the remainder when  is divided by 6?

Go back to the divisibility concepts discussed. When  balls are split into groups of 6, we will have 2 balls leftover. If we are given 1 more ball, it will join the 2 balls and now we will have 3 balls leftover. The remainder will be 3.

What happens in the case of  – what will be the remainder when this is divided by 6? This additional 6 balls will just make an extra group of 6, so we will still have 2 balls leftover.

What about the case of ? Now, of the extra 9 balls, we will make one group of 6 and will have 3 balls leftover. These 3 balls will join the 2 balls leftover from , giving us a remainder of 5.

Now, what about the case of ? Recall that . The number of groups will double and so will the remainder, so  will give us a remainder of .

On the other hand, if  gives us a remainder of 4 when divided by 6, then  divided by 6 will have a remainder of , which gives us a remainder of 2 (since another group of 6 will be formed from the 8 balls).

Let’s consider the tricky case of  now. If  gives us a remainder of 2 when it is divided by 6, it means:

Note here that the first and the second terms are divisible by 6. The remainder when you divide this by 6 will be 4.

We hope you understand how to deal with these various cases of remainders. Let’s take a look at a GMAT sample question now:

Question: If  is a positive integer and  is the remainder when  is divided by 8, what is the value of ?

Statement 1: When  is divided by 8, the remainder is 4.

Statement 2: When  is divided by 8, the remainder is 2.

This is not our typical, “When  is divided by 8,  is the remainder” type of question. Instead, we are given a quadratic equation in the form of  that, when divided by 8, gives us a remainder of . We need to find . This question might feel complicated, but look at the statements – at least one of them gives us data on a quadratic! Looks promising!

Statement 1: When  is divided by 8, the remainder is 4

We know that when  is divided by 8, the remainder is 4. So no matter what  is, , when divided by 8, will only give us a remainder of 4 ( is a multiple of 8, so will give remainder 0).

 when divided by 8, gives remainder 4. This means  is divisible by 8 and would give remainder 0, further implying that  would be 1 less than a multiple of 8, and hence, would give us a remainder of 7 when divided by 8. This statement alone is sufficient.

Let’s look at the second statement:

Statement 2: When  is divided by 8, the remainder is 2

When  is divided by 8, the remainder is 1. When  is divided by 8, the remainder is 2. So when  is divided by 8 the remainder will be 1+2 = 3.

When  is divided by 8, remainder will be 3 + 4 = 7. This statement alone is also sufficient. Because both statements alone are sufficient, our answer is (D).