Our Divisibility and Remainders module 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 “x” gives you a remainder of 2 when divided by 6. What will be the remainder when x + 1 is divided by 6?
Go back to the divisibility concepts discussed. When x 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 x + 6 – 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 x + 9? 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 x, giving us a remainder of 5.
Now, what about the case of 2x? Recall that 2x = x + x. The number of groups will double and so will the remainder, so 2x will give us a remainder of 2*2 = 4.
On the other hand, if x gives us a remainder of 4 when divided by 6, then 2x divided by 6 will have a remainder of 2*4 = 8, 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 x^2 now. If x gives us a remainder of 2 when it is divided by 6, it means:
x = 6Q + 2
x^2 = (6Q + 2)*(6Q + 2) = 36Q^2 + 24Q + 4
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 z is a positive integer and r is the remainder when z^2 + 2z + 4 is divided by 8, what is the value of r?
Statement 1: When (z−3)^2 is divided by 8, the remainder is 4.
Statement 2: When 2z is divided by 8, the remainder is 2.
This is not our typical, “When z is divided by 8, r is the remainder” type of question. Instead, we are given a quadratic equation in the form of z that, when divided by 8, gives us a remainder of r. We need to find r. 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 (z−3)^2 is divided by 8, the remainder is 4
(z – 3)^2 = z^2 – 6z + 9
We know that when z^2 – 6z + 9 is divided by 8, the remainder is 4. So no matter what z is, z^2 – 6z + 9 + 8z, when divided by 8, will only give us a remainder of 4 (8z is a multiple of 8, so will give remainder 0).
z^2 – 6z + 9 + 8z = z^2 + 2z + 9
z^2 + 2z + 9 when divided by 8, gives remainder 4. This means z^2 + 2z + 5 is divisible by 8 and would give remainder 0, further implying that z^2 + 2z + 4 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 2z is divided by 8, the remainder is 2
2z = 8a + 2
z = 4a + 1
z^2 = (4a + 1)^2 = 16a^2 + 8a + 1
When z^2 is divided by 8, the remainder is 1. When 2z is divided by 8, the remainder is 2. So when z^2 + 2z is divided by 8 the remainder will be 1+2 = 3.
When z^2 + 2z + 4 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.
Answer (D)
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