When the positive integer A is divided by 5 and 7, the

Question

When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

Bunuel · Accepted Answer

When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively: A=5q+3 (A could be 3, 8, 13, 18 , 23, ...) and A=7p+4 (A could be 4, 11, 18 , 25, ...). There is a way to derive general formula based on above two statements: Divisor will be the least common multiple of above two divisors 5 and 7, hence 35 . Remainder will be the first common integer in above two patterns, hence 18 --> so, to satisfy both this conditions A must be of a type A=35m+18 (18, 53, 88, ...); The same for B (as the same info is given about B): B=35n+18 ; A-B=(35m+18)-(35n+18)=35(m-n) --> thus A-B must be a multiple of 35. Answer: C. More about this concept: manhattan-remainder-problem-93752.html?hilit=derive#p721341 good-problem-90442.html?hilit=derive#p722552 Hope it helps.

KarishmaB · Answer

If I have a number n which when divided by 5 gives a remainder 3 and when divided by 7 gives a remainder 4, the number is of the form:
n = 5a + 3 
n = 7b + 4 
I will need to check for the smallest such number.
I put b = 1. n = 11. Is it of the form 5a + 3? No.
Put b = 2. n = 18. Is it of the form 5a + 3? Yes.

When 18 is divided by 5, it gives a remainder of 3. When it is divided by 7, it gives a remainder if 4.

Next such number will be 35 + 18 because 35 will be divisible by 5 as well as 7 and whatever is the remainder from 18, will still be the remainder
Next will be 35*2 + 18
and so on...
Difference between such numbers will be a multiple of 35 so your answer is 35.
 Note: Actually, because of this reasoning, you just had to take the LCM. You didn't even need to find the first such number!

I have discussed this topic a little more in detail here:  https://gmatclub.com/forum/good-problem-90442-20.html#p814507

SVaidyaraman · Answer

The easiest way to approach these problems is by taking an example

1. The first choice is 18. Take it as B
2. The next choice is 53. Take it as A
3. A-B=35

c is the only choice that is correct.

gracie · Answer

if A and B, when divided by the same divisors, leave the same remainders,
then the difference between them will always be a multiple of the product of those divisors
5*7=35
C

MvArrow · Answer

I chose the same approach but only a further tip to speed up: When you found the value of A you can simply try to add every value from the answer choices: if the number obtained satisfy the initial divisibility conditions then you have found the correct answer

BigUD94 · Answer

VeritasKarishma

Mam, I did it this way, is this a correct way?

A=5Q+3 & A=7s+4
so equating we get

5Q=7s+1

Which is for both A&B so the numbers derived are 18 and 53 solving for the above, hence the factor of 53-18= 35

C

KarishmaB · Answer

It seems that basically you have found 2 numbers which satisfy both conditions and taken their difference (which would need to be a multiple of at least one of the choices). It is correct. I hope you understand why it is correct. It you are not sure, check out the link I have given above in my post.

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When the positive integer A is divided by 5 and 7, the

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