Manhattan GMAT Challenge Problem of the Week – 11 May 2010

by , May 11, 2010

Welcome back to this weeks Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!

Question

Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that n leaves after division by 30?

(A)3

(B)12

(C)18

(D)22

(E)28

Answer

The simplest way to approach this problem is to work backwards from the answer choices. Lets construct a possible value of n for each choice, and then test those values against the given constraints.

Since we are asked for the remainder after division by 30, the easiest possible value of n for each choice is 30 more than the choice.

(A) 3 + 30 gives us n = 33

(B) 12 + 30 gives us n = 42

(C) 18 + 30 gives us n = 48

(D) 22 + 30 gives us n = 52

(E) 28 + 30 gives us n = 58

Now test those values of n against the constraints.

(A) n = 33 divided by 6 gives remainder 3 FAIL

(B) n = 42 divided by 6 gives remainder 0 FAIL

(C) n = 48 divided by 6 gives remainder 0 FAIL

(D) n = 52 divided by 6 gives remainder 4 PASS

(E) n = 58 divided by 6 gives remainder 4 PASS

We can now just test the surviving choices for how they behave upon division by 5. To leave remainder 3 after division by 5, a number must end in either 3 or 5:

(D) n = 52 divided by 5 gives remainder 2 FAIL

(E) n = 58 divided by 5 gives remainder 3 PASS

The correct answer is therefore (E).

Another way to approach this problem is to translate the given language of remainders into the language of multiples. If n leaves a remainder of 4 after division by 6, then n is 4 more than a multiple of 6. Leaving aside the size requirement for a moment, we can see that n could be 4, 10, 16, 22, 28, 34, etc.

Likewise, if n leaves a remainder of 3 after division by 5, then n is 3 more than a multiple of 5. Again leaving aside the size requirement, we can see that n could be 3, 8, 13, 18, 23, 28, 33, etc. As we noted earlier, n must end in 3 or 8.

We might now spot 28 on both lists. Although n is not actually allowed to be 28 (because n must be larger than 30), we might try adding 30 to it to get 58. Since 30 is a multiple of 6, adding 30 to 28 wont change the fact that after division by 6, well get 4 as the remainder. The same idea holds true for 5: since 30 is a multiple of 5, adding 30 to 28 wont change the fact that after division by 5, well get 3 as the remainder. This way, we have constructed a possible n without using the answer choices.

Finally, the remainder after dividing 58 by 30 is 28.

Again, the correct answer is E.

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