What is the greatest 6-digit number when divided by?

Question

What is the greatest 6-digit number when divided by 6, 7 ,8 , 9, and 10 leaves a remainder of 4, 5, 6, 7, and 8 respectively?

A. 456780
B. 678910
C. 997479
D. 997918
E. 997920

IanStewart · Accepted Answer

When you divide a positive integer by 10, the remainder will just be the units digit. We know the remainder is 8 when we divide by 10, so D is the only possible answer.

Or you could use the remainder by 9 -- if you want to know a large number's remainder when you divide by 9, you can sum its digits, and then take the remainder of that sum when you divide by 9. Only answer D gives you a sum of digits with a remainder of 7 when you divide the sum by 9.

Princ · Answer

OA: D

Options A, B and E are divisible by 10 i.e remainder is Zero.
C gives a remainder of 9 when divided by 10.
Only D gives a remainder of 8 when divided by 10.

mikemcgarry · Answer

Dear sandeepk123 , I'm happy to respond. :-)

My friend, it is very hard to write truly GMAT-like Quant questions. It's true that in a good GMAT question, it may look very complicated but involve an easy shortcut, as this one does. The shortcut here, though, is a bit too easy: the remainder when dividing by 10 has to be the units digit. Once one sees the shortcut, there is absolutely no work, and I imagine a large number of students will see this shortcut. My friend, I don't know if you are familiar with the idea of Item Response Theory . This is the science of what constitutes a truly high quality test question. The psychometricians who design the GMAT are experts in Item Response Theory. I only know a little about it myself. The gist is that a well-written question really strikes this delicate balance: many students find the question challenging, or are persuaded by incorrect trap answers, and only high performing students choose the correct answer. This question has an obvious right answer, but there is really no inducement for anyone, other than a blind guesser, to choose any of the other four answers. The questions would likely show similar performance among high performers and low performers. Thus, it is not an item that discriminates between users well. There are many layers involved in writing a truly GMAT-worthy math question. It's not just a matter of being clever with the math itself. Does all this make sense? Mike

MathRevolution · Answer

There is a relation between divisors and remainder, which is remainders is 2 less than divisor, that is,  remainder = divisor - 2 .

And the answer should be  a common multiple of ( 6, 7, 8, 9, 10 )  - 2 as follows.

Let  L  be the  LCM(6,7,8,9,10) . Then  L = 2^3 3^2 5^1 7^1 = 2520 .

2,520 * 400 - 2,520 * 4 = 1,008,000 - 10,080 = 997,920

Therefore, the answer is  997,920 - 2 = 997,918 .

The correct choice is D

ManifestDreamMBA · Answer

Multiple ways to do this but easiest is to spot a 5 or 10 in the divisors. 
5 can have only 2 options for unit's place. E.g. remainder of 1 can be a result of dividend ending in 1 or 6, 2 for 2 or 7 and so on.
10 is even easier, since it just spits out the unit's place as reminder whatever that might be.
Given the remainder when divided by 10 leaves a remainder of 8, the dividend must end in 8, D is the answer

What is the greatest 6-digit number when divided by?

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What is the greatest 6-digit number when divided by?

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