When the positive integer A is divided by 5 and 7

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

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.
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Although there are methods.. this particular problem does not require any methods..

A divided by 5,7 gives 3 and 4
B divided by 5,7 gives 3 and 4.

means A and B are equidistant from the multiples of 5,7. the LCM of 5 and 7 is 35, so numbers like A & B repeat after every 35 numbers. so 35 is the answer.

With the remainders remaining same for both the required integers, it means there difference should be divisible by both 5 & 7

Out of all the five given options, only 35 stands out

Answer = C

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.


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


By the conditions we may put A=5K+3 = 7L+4, B=5M+3 = 7N+4. So we have A-B= 5K+3-5M-3 = 5*(K-M)= 7L+4 - 7N-4=7*(L-N).

We may conclude that A-B is divisible by 5 and 7. That means 35 is a factor of A-B.

So the answer is (C).
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because the difference between any values of A and B
will always be a multiple of the product of their common divisors, 5 and 7,
the correct answer is 35
C

Since when A is divided by 5, the remainder is 3, A can be values such as:

3, 8, 13, 18, ...

Since when A is divided by 7, the remainder is 4, A can be values such as:

4, 11, 18, ...

We see that the smallest number A can be is 18; the next value for A is 18 plus the LCM of 5 and 7, that is, A = 18 + 35 = 53. And we can keep adding 35 to obtain succeeding values of A. That is, A can be values such as:

18, 53, 88, 123, …

Since B has the same condition as A, then B can be any of the values above. Therefore, the difference between A and B must be a multiple of 35 (for example, if A = 88 and B = 18, A - B = 70 = 2 x 35).

Alternate Solution:

Since both A and B produce the same remainder when divided by 5, A - B must be divisible by 5. (The reason is the following: Since A produces a remainder of 3 when divided by 5, A must be expressible as 5p + 3 for some integer p. Since B produces a remainder of 3 when divided by 5, B must be expressible as 5q + 3 for some integer q. Then, A - B can be expressed as 5p + 3 - (5q + 3) = 5p - 5q = 5(p - q); which is a multiple of 5)

Similarly, since both A and B produce the same remainder when divided by 7, A - B must be divisible by 7. Since A - B is divisible by both 5 and 7, it must be divisible by LCM of 5 and 7, which is 35.

Answer: C
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I really like this method mentioned above, but myself used another approach, more time consuming.
From given information, we know that A must be 18 (at least) to satisfy both conditions.
Likewise, we know B must be 53( at least) to satisfy both conditions.
So, we have 53-18=35 (C)

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