Problem Solving

If j and k are positive integers, j - 2 is divisible by 4 and k - 5 is divisible by 4, all of the following could be the value of j - k EXCEPT:

a)43
b)33
c)21
d)13
e)5

I can understand from the q that since j-2/4 is divisible by 4 j is even & k-5is divisible by 4 k is odd, so difference will be odd but how come it can not be 43 ?

A

J-2=4a

J=4a+2

K-5=4b

K=4b+5

J-K=4(a-b)-3

As you can see from the option,Except A others can be derived from the above equation by sub value (a-b) as 2,4,6 and 9.

Hope this clarify!!

When a is divided by b to get a quotient q and remainder r, we can write:

a = bq + r

Now, j -2 is divible by 4, so remainder = 0

We can setup this equation -> j - 2 = 4p

Similarly, k - 5 = 4q

where p and q are some quotients

subtracting, we get

j - k + 3 = 4(p-q)

=> j - k = 4(p-q) - 3

Which means the difference of j and k should be 3 less than a multiple of 4

Only 43 satisfies this criteria, hence (A)

KMittal, you are absolutely correct. But at the end I think you meant to say that 43 is the only number that does NOT satisfy the criteria.

The trick to doing these problems is to play with the information given, until you end up with the variables stated in the problem (in this case, j - k) just as selango and KMittal have done.

Use variables to represent the factors of that result in j-2 and k-5: j-2 = 4x and k-5=4y.

Then isolate both j and k; j = 4x + 2 and k = 4y + 5, and subtract:

j - k = (4x + 2) - (4y + 5)

Simplify:

j - k = 4(x-y) - 3

This means that (j-k) can be any multiple of 4 subtracted by 3. All of the answer choices are multiples of 4 subtracted by 3 except for (A), which is 43 and 46 is not evenly divisible by 4.

Hey guys,

Great discussion - if anyone is looking for another, less-algebraic way to go through this one, I think it fits pretty nicely into the "find the pattern" style of Number Properties questions.

Because they're looking for the difference between two sequences, there will be a pattern in that difference, and if you test it with a few smaller numbers you can likely find the pattern and extrapolate it to the answers. Let's look at a few possible values of j and k:

j - 2 is divisible by 4, so j could be:

6, 10, 14, 18, 22, 26...

k - 5 is divisible by 9, so k could be:

5, 9, 13, 17, 21, 25...

Now, we're looking for the difference j-k, and all of the answer choices are positive, so let's take a high value of j and start subtracting possible values of k. Call j 26 and subtract:

j - k =
26 - 25 = 1
26 - 21 = 5
26 - 17 = 9
26 - 13 = 13

If you look at the differences, they themselves form a pattern: 1, 5, 9, 13. You can extrapolate that to include 17, 21, etc., and when you ask yourself why the pattern holds, you'll probably see that because both sequences for possible j and k values increase each time by 4, so will the difference between them.

For any number in that set of possible differences, we can just add a multiple of 4 and find another possible difference. We've already covered 5, 13, and 21 on our list, and 33 is just 21 + 12 (a multiple of 4), so we can eliminate those. 43 is 10 bigger than 33 - it's not on the the same cycle, so it's not a possible difference.


What I like most about that "find the pattern" strategy is that, if nothing else, it gets you thinking about the values instead of scrambling to try to set up algebra. Even if the algebra ends up being a little quicker, by plotting a few values in a case like this you'll likely be able to notice the algebra and convert over. And if you've ever found yourself staring at a question with nervous energy hoping that you could find a way to spark some action, this method works pretty well to at least get you moving in a positive direction.