When a positive integer is divided by 4, the remainder is r; when divi

\(n=4p+r\)
\(n=9q+R\)

the remainder is always smaller then the dividor, so r<4, and R<9.
Since the remainder is always an integer, the greatest possible value for r and R is: r=3, R=8

\(r^2+R=9+8=17\)

Using the concept of divisibility, you can get the answer with minimum calculation.

When a number is divided by 4, the maximum value of r can be 3.
When a number is divided by 9, the maximum value of R can be 8.

Now all we need to figure out is whether we can have a number such that when divided by 4, it gives remainder 3 and when divided by 9, it gives remainder 8.

Imagine the number split into groups of 9 and 8 leftover. When the same is divided by 4, the 8 leftover will be evenly split into groups of 4. From each group of 9, we will make 2 groups of 4 each such that 1 is leftover from each group of 9. We want 3 to be leftover when we divide by 4 and this will be possible if we have 3 groups of 9.

So basically such a number could be 3*9 + 8 = 35 etc

Hence, maximum value of r^2 + R = 3^2 + 8 = 17

Answer (C)

Is this a real 600 level question? :O

No. About 680 - 700 level I would say.

You can check difficulty level of a question along with the stats on it in the first post. For this question Difficulty: 600-700 Level. The difficulty level of a question is calculated automatically based on the timer stats from the users which attempted the question.

Another way that works is to gauge the LCM of the two divisors (4 and 9) and the LCM - 1 will yield that number (\(36-1=35\)) for which both divisions result in the maximum remainder (3 and 8 respectively).

I don´t actually know why this works (probably is based on the theory that some of the other users have already posted) but if you encounter something similar, it´s a ready-to-apply and go.


-

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.