If A and B are single digit non-zero numbers, what is the product of A

Question

If A and B are single digit non-zero numbers, what is the product of A and B?

(1) When the two-digit number AB is added to BA, the digits at units, tens and hundreds place in the sum are L, 8 and 1 respectively.
(2) When the two-digit number AB is added to BA, the digits at units place and tens place in the sum differ by 1.

IanStewart · Accepted Answer

It's a bit of a strange question, and there are a few ways to break it down, but inspection works well. If, from Statement 1, the two numbers add up to 180-something, one of the numbers must be 90 or greater (their average is greater than 90, and the two numbers can't both be below average). So one of our digits is 9. But if the other digit was 7, our sum wouldn't be big enough, and if the other digit was 9, the sum would be too large, so our digits must be 9 and 8, and their product is 72, so Statement 1 is sufficient.

Statement 2 will be true any time we have to carry 1 when we add, so our numbers could be 65 and 56, say, or 85 and 58, among other possibilities, and the answer is A.

You can also write things algebraically: the two-digit number AB is equal to 10A + B. So the sum in Statement 1 is equal to 10A + B + 10B + A = 11A + 11B = 11(A + B), and this is a multiple of 11. If that equals some number in the 180s, it must equal 187, the only multiple of 11 between 180 and 189, so 11(A+B) = 187, and A+B = 17. Since A and B are single-digit integers, they can only be 8 and 9. So Statement 1 is sufficient.

Bunuel · Answer

This question is a part of  Are You Up For the Challenge: 700 Level Questions  collection.

andrenoble · Answer

I think it's a very good question. That's how I approached it:

Statement I

I basically says that AB+BA = 18L (meaning some number from 180-189). It's pretty easy to "plug in" numbers to see what matches what is given in the question:
1) 97 + 79 = 176
2) 98 + 89 = 187
3) 99 + 99 = 198

As we can see, only 98+89 gives the required sum.  Sufficient

Statement II

A + B = (hundreds?)(tens)(units); |tens - units| = 1

Two important things:
1) We do not know if there are hundreds or not.
2) Difference of units and tens is 1, but we do know which is greater (it doesn't really matter here, but wanted to outline it anyway). There are numerous possible solutions that match, e.g., 
1) 97 + 79 = 176
2) 98 + 89 = 187 ->  insufficient

A

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If A and B are single digit non-zero numbers, what is the product of A

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GMAT Data Sufficiency (DS) Questions

If A and B are single digit non-zero numbers, what is the product of A

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