GMAT question

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ironsheep

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This is GMAT question 98 from "Gmat official Guide 2019 Quantitative Review"--- "For positive integers a and b, the remainder when a divided by b is equal to the remainder when b is divided by a. Which of the following could be a value of ab?
1. 24
2. 30
3. 36

The answer is 36 only.

My answer is that you can't figure out this problem. 2/4 equals 1/2 with no remainder and 4/2 is 2 with no remainder. 3/6 is 1/2 and 6/3 is 2, both with no remainder. 2 times 4 is eight and 3 times 6 is 18. Not enough information
 
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You aren't doing remainders correctly. For your example of 2 and 4, 2 ÷ 4 = 0 r 2, and 4 ÷ 2 = 2 r 0. The remainders are not the same.

I think they are going for the idea that the only way it could happen is if a = b, so ab has to be a square. For example, if a = b = 6, then 6 ÷ 6 = 1 r 0 both ways, and the remainders are the same.

To prove that, suppose that a > b. Then the remainder of a ÷ b has to be less than b, while the remainder of b ÷ a is just b. The same sort of thing will happen if a < b. But they aren't asking for a proof.

I imagine a lot of people taking this test might not have done elementary arithmetic with remainders for a long time.
 
What makes you think a = b, how did you come to that conclusion?? Your right about the remainders, but the only way to find that out is to use the division box thing with 4 outside the box and 2 inside.
 
2 divided by 4 does have a remainder, namely 2. Your example is fallacious.
 
ironsheep said:
What makes you think a = b, how did you come to that conclusion?? Your right about the remainders, but the only way to find that out is to use the division box thing with 4 outside the box and 2 inside.
Click to expand...
I explained my thinking:
Dr.Peterson said:
To prove that, suppose that a > b. Then the remainder of a ÷ b has to be less than b, while the remainder of b ÷ a is just b. The same sort of thing will happen if a < b.
Click to expand...
In other words, the remainder has to be less than the divisor. But it can't be less than both numbers, unless it's zero!

I'm not sure this is a good problem to put on a test like this, for several reasons.
 
I am confused. I get the remainder has to be the same, but what is a and what is b? How do you find that out with the three numbers they give you??
 
It is a tricky question. You cannot and need not find numeric values for a and b. Once you know, however, that a = b = some unknown integer, you know that ab = a^2 and is a perfect square. Only 36 is a perfect square out of the choices given. So 36 could be an answer, but the other two could not be valid answers. The question does not ask what the answer is, but what it could be.

There is not enough information to determine a unique answer, but there is enough information to determine what numbers cannot be valid answers.
 
Wow, I wouldn't have stopped to think about whether or not a=b, so on the GMAT exam, I would never have gotten any chance of getting problems similar to this correct, but how did you come to suspect that a might equal to b??
 
I don't know how Dr. P attacked it, but I was working from

[MATH]a = bk + r, \ b = am + r, \ 0 \le r < a,\text { and } r < b.[/MATH]
That is, I was thinking about the definition of remainders. Dr. P may have had a cleverer way to do it because he wrote his answers while I was still working on mine.
 
JeffM said:
I don't know how Dr. P attacked it, but I was working from

[MATH]a = bk + r, \ b = am + r, \ 0 \le r < a,\text { and } r < b.[/MATH]
That is, I was thinking about the definition of remainders. Dr. P may have had a cleverer way to do it because he wrote his answers while I was still working on mine.
Click to expand...


I am still confused, also on the GMAT I won't be able to spend alot of time on the each question, so any tricks or rules?? What is with the 0<r<a
 
I have no idea how they expect a typical taker of their test to think about a problem like this; that's why I don't like it. A mathematician will approach it very differently from someone else -- likely making it harder than it needs to be. I assumed this is not meant to be approached as a proof problem, but in terms of elementary understanding of arithmetic, so I thought at that level: start with an example, to see what you can learn. I quickly saw that it would work if a=b, so that 36 would be a possibility (6=6); I was left to convince myself that a=b was the only possibility. Your example of 2 and 4, when corrected, reminded me that a remainder has to be less than the divisor, which showed that if a and b were different, the remainders couldn't be the same.

One "trick" they may want you to learn from this is that you can never assume that a and b are not equal. Perhaps that is part of the reasoning skills they consider important. I doubt that the concept of remainder is a central concept they care about.

Another is to read carefully. They didn't imply that 36 was the only possible value of ab, or that you could know what a and b are. They very carefully asked, "Which of the following could be a value of ab?". Always look at exactly what they ask.
 
Okay, thank you both for making this problem a little clearer as I couldn't understand when the author tried to explain it in the book.
 
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