If M and N are positive integers that have remainders of 1

Question

If M and N are positive integers that have remainders of 1 and 3, respectively, when divided by 6, which of the following could NOT be a possible value of M+N?

(A) 86
(B) 52
(C) 34
(D) 28
(E) 10

Bunuel · Accepted Answer

M = 6p + 1, where p is integer >=0, so M can be 1, 7, 13, etc.
N = 6q + 3, where q is integer >=0, so N can be 3, 9, 15, etc.

M + N = 6(p + q)+4, hence M + N is multiple of 6 plus 4 = 10, 16, 22, 28, 34, etc.

Only answer which is not of this type is 86, 86 = 13*6 +  2 .

Answer: A.

Hope it's clear.

Madsushi · Answer

Assume M = 6a+1 and N = 6b+3
M+N = 6(a+b)+4
Hence answer should be such that when subtracted by 4, it should be divisible by 6.
86-4 = 82 is not divisible by 6. Hence A is the answer.

sagarsabnis · Answer

but still can you give me the the value of M and N which will add up to 10. If you check the other way round the least numbers are 7 and 9 which adds up to 16 so how come 10 is possible?

Bunuel · Answer

The problem in your reasoning is that you are not considering the values of M and N when, p and q are 0. If you do you'll get the least values: M=1 and N=3. So if you take the value of M when p=1 and value of N when q=0 you'll get:

M=7 and N=3 --> M+N=10.

sagarsabnis · Answer

Yeah now i do get it...Thanks mate!!!

nsvarunns · Answer

I did it this way
M=6p+1
N=6q+1

We need M+N=6p+1+6q+3=6(p+q)+4

Pick numbers for p & q
Since it is an addition of p & q and the answer to this expression should be an integer (because all the numbers being added are integers),
we just need to choose values so that we get integer multiples of 6 so
p+q=0 ; M+n = 4
p+q=1 ; M+N = 10
P+q=2 ; M+N = 16

and so on, so basically you get something like - 4,10,16,22,28,34,..... all the other options were turning up.

Then I directly tried p+q=12 because it was closer to 86 for the first option, i got 76, then tried with 13 - got 84, 14 got 90.. no 86.

So Answer A.

Is the approach correct?

KarishmaB · Answer

Your approach was not wrong but a little cumbersome. Instead of picking values for p and q and trying to get to the options, pick the options and find out whether they suit this format.

52 = 6*8 + 4 so p+q = 8. Hence 52 can be the value of M+N
Similarly check for other options. At every step, you get closer to the solution else you could end up waiting for a long time before you get 4 of the 5 options.

mindmind · Answer

M & N are divided by 6 : R =1 & 3 resp

M+ N divided by 6 : R 1+ 3 = 4

So divide the options by 6 .. & R should be 4.

Only ..86 has a remiander of 2 .....instead of 4..

Answer A

chiccufrazer1 · Answer

@bunuel..why are we considering the integers 1 and 3 in the list of integers when the stem question says that when each of the numbers is divided by six,the remainder is 1 and 3 respectively..when 1 is divided by six am sure we would not get 1 as a remainder neither would we get 3 as a remainder when the integer 3 is divided by 6

Posted from my mobile device

Bunuel · Answer

That's not correct. When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer. For example, 7 divided by 11 has the quotient 0 and the remainder 7 since 7=11*0+7 Hence, 1 divided by 6 yields the remainder of 1 and 3 divided by 6 yields the remainder of 3. For more check Remainders chapter of Math Book: https://gmatclub.com/forum/remainders-144665.html

bumpbot · Answer

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If M and N are positive integers that have remainders of 1

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