In the above equation, each of the letters A, B, C, D, E, F, G, H, I a

Question

Capture.PNG In the above equation, each of the letters A, B, C, D, E, F, G, H, I and J represents a distinct digit from 0 to 9. What is the remainder when maximum possible value of 'HIJ' is divided by 9. {'HIJ' is a 3-digit number} A. 0 B. 1 C. 3 D. 7 E. 9

IanStewart · Accepted Answer

A couple of ways to do this problem: you can work out what the maximum value of HIJ is equal to. H clearly needs to be 9, and E needs to be 8. If you try to make HIJ equal 970-something, you quickly find the remaining digits aren't large enough to get you there (the options you have left for the tens digits A, C, and F are 4, 5 and 6, which don't add up to 17). But if you then try to make HIJ into 960-something, you find that's possible in one way - you just want to make the units digits 0, 1 and 2. So our sum could be something like 40+51+872 = 963, for example. Now we can answer the question - the remainder when we divide 963 by 9 is zero.

If you know remainder theory (modular arithmetic) really well, you can answer the question in a different way, but this isn't really something you'd need on the GMAT except in much simpler cases. You'd need to know two things:

- when you're dividing specifically by 9, and want to know the remainder you'll get, you can just sum the digits of your number and take the remainder by 9. So if you divide 317 by 9, for example, the remainder will be 2, because the remainder is 2 when we divide 3+1+7 = 11 by 9.

- when you add two or more numbers, and want a remainder when you divide their sum by 9 (or by anything else) you can first find the remainder of each number by 9 (or by whatever other number you're working with) and add those remainders together (taking a remainder again if the answer is too large). So if you want the remainder when you divide 317 + 899 by 9, you can find the remainders 317 and 899 give you, which is easy by summing digits - we get 2 and 8 - and just add those to get 10. That's too big to be a remainder by 9, so we take a remainder by 9 again to get the answer, 1.

So using all that theory, in this question, the remainder R we get when we divide AB + CD + EFG by 9 will just be equal to the remainder we get when we divide A + B + C + D + E + F + G by 9. Say we call the sum of those seven digits "S". Then S = 9q + R for some quotient q. Because remainders add, H+I+J will also give a remainder R when we divide by 9. If we call the sum of those three digits "x", then x = 9m + R, for some quotient m. But we know S + x is the sum of all ten digits from 0 through 9, so S + x = 45. So

9q + R + 9m + R = 45
2R = 45 - 9(q + m)

and since this is an equation of integers, and 9 is a factor of the right side, it must also be a factor of the left side, and thus R must be a multiple of 9. Since R is less than 9, R can only be zero.

Using a method like that might be easy if you have a ton of experience with it, but most GMAT test takers won't (and won't need it), so I think the first method above will be the most practical one for most test takers to use. The one advantage of the second method is that it shows the answer is zero even when we don't maximize HIJ (that's not a detail in the question that actually affects the answer).

EgmatQuantExpert · Answer

Solution

Given

• A through J, each alphabet represents a separate digit from 0 through 9.
• AB + CD + EFG = HIJ

To Find 
 • The remainder when HIJ is divided by 9.

Approach and Working Out

We need to use the concept of digit sum here.
Let’s take a few examples -

• Case 1 : 4 + 5 = 9
 Sum of digits on LHS = 9
Sum of digits on RHS = 9 
 o The sum is same 
• Case 2 : 4 + 7 = 11
 Sum of digits on LHS = 11
Sum of digits on RHS = 2 
 o The reduction is due to carry and writing 10 as 1 which results in reduction of 9. 
• Case 3 : 5 + 7 + 9 = 21
 Sum of digits on LHS = 21
Sum of digits on RHS = 3 
 o The reduction is due to carry and writing 20 as 2.

So we can conclude that, when we carry ‘1’ then the sum of digit decreases by 9 and when we 2 carry the sum decreases by 18 and so on.

• Sum of digits of A + B + C …. J = 0 + 1 + 2 … 9 (not necessarily in the same order) = 45.
• We can say the sum of digits for A + B + C + D + E +F +G must be greater than HIJ by 9 or 18 or 27.
• Since the sum of digits on LHS and RHS is a multiple of 9(45) and the difference of LHS and RHS is also a multiple of 9. 
  o We can say that A + B + C +D + E + F + G and H + I + J will be a multiple of 9.  
 • Hence, HIJ, the three-digit number, is a multiple of 9 and our answer is 0.

Correct Answer: Option A

mSKR · Answer

usually answer of such kind of questions that need lot of guesses if need to define the values.
Best guess of such answers is 0?

Could this question be changed in such a way that remainder is not zero and hence question can not be solved without proper method.

please suggest  IanStewart   EgmatQuantExpert

IanStewart · Answer

Yes, it's easy to change the problem to make the answer nonzero, just by changing the set of digits you're allowed to use. To give an extremely simple example, if you had the equation A+B = C, where A, B and C are distinct digits between 1 and 3 inclusive, the remainder is not 0 when we divide C by 9 (since the equation must be 1+2 = 3). You could modify the problem above in a similar way, but I won't do that, because you can't just do it randomly, so it would take some effort -- many equations one might invent won't have any solutions at all if there are extreme restrictions on the set of digits.

But there really isn't any reason to worry about how a problem already slightly too hard for the GMAT could be modified to become even harder. As for how to guess at a question like this one, there isn't going to be a strategy statistically better than random guessing, once you rule out illogical answers. Here, since we need a remainder when we divide by 9, we can instantly rule out answer E, because "9" can never be a remainder when you divide by 9. But if this were a real GMAT question, once you ruled out that one impossible answer, you'll do just as well in the long run by guessing A as by guessing D. So if you want to guess 'zero' at questions like this, there'd be no harm in doing that, but it's also not likely to help.

EgmatQuantExpert · Answer

To change the answer from 0 to anything else it can be done in the following ways (not exhaustive).

I. By changing the set of numbers as mentioned by Stweart. 
II. Changing the divisor. 
III. Another way may be to change the question and ask the possible sum of H + I + J.

This will make the approach a little more cumbersome. That is out of the scope of GMAT.

bumpbot · Answer

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