Question: K Divided by 67

Comment on K Divided by 67

HI, I'm struck in choosing the two methods for solving the problem. At times method-1 seems easy and sometimes method 2 seems easy. It is difficult when I take long time in solving problem tro method 1, where it can be solved easily by method-2. How can I identify the best way for choosing the methods.

gmat-admin's picture

This question can be applied to most GMAT math questions in which there is more than one approach.

If you can identify two approaches, it helps to get a general feeling for the number of steps/calculations each approach will take, and then choose the faster one. Yes, easier said that done :-)

That said, when it comes to Integer Properties questions, I find that the best strategy is typically choosing a number that satisfies the given information and going from there.

Hi Sir, understood first option clearly but stuck with option 2, how did we consider 89 as possible value of K, since 89<2345. Please guide.
gmat-admin's picture

When it comes to remainders, we have a nice rule that says:

If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.

For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

More here:

gmat-admin's picture

Here's another way to think of it:

89 is EITHER divisible by 2345 OR not divisible by 2345. One of those two things must be true.

Since it's clear that 89 is not divisible by 2345, then there must be a remainder when we take 89 and divide it by 2345.

89 divided by 2345 equals 0 with remainder 89.

Likewise, 17 divided by 9 equals 1 with remainder 8

And 21 divided by 4 equals 5 with remainder 1

for the second approach, don't I need to check the other values of K not just the 89? Like, is there a chance to get a different remainder if I test the other values of K?
gmat-admin's picture

Great question, Mohammad!

IF one of the answer choices were "Cannot be determined," then it would be advisable to test other values.

However, since all 5 answer choices are numbers, we can conclude that, no matter what value we test, we will always get the same answer.

So, for example, it cannot be the case that the correct answer is D when we test k = 89, and the correct answer is something other than D when we test k = 2434.

Does that help?



Hi Brent,

The second approach seems to me to be incomplete. What if we had a divisor of the original number that was not divisible by 67, say instead of 2345 we had 2346 (not divisible by 67), so then if we were to just pick first value 89, then the answer would have been wrong. Don't we need to check first if the divisor is divisible into second number first, if it is than we can proceed with picking smallest possible value of K, if it is not divisible than we need to pick say second possible value of K that contains original divisor (in my example 2346+89) and divide that number by 67.
gmat-admin's picture

Great observation!!!!

You're absolutely correct; the 2nd approach hinges on the fact that 2345 is divisible by 67.

That said, when we test 89 as a possible value of k, we find that remainder MATCHES one of the answer choices, which tells us that 22 must be ONLY possible remainder when k is divided by 67, regardless of which possible k-value we choose to test (otherwise, the question would have more than one correct answer).


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