A number 'n' when divided by 5 leaves remainder 1
and when divided by 7 leaves remainder 3.

The number will be of the form '7A + 3'

A=1 -> 10 (when divided by 5 remainder = 0)
A=2 -> 17 (remainder 2)
A=3 -> 24 (reminder 4)
A=4 -> 31 (reminder 1) - - - - - Satisfies our condition

Now its said when 'k' is added to the number, its a multiple of 35.

So, 4 should be added to 31 to make it a multiple of 35. So k=4 IMO

in this kind of problems, we should translate the problem into mathematical equation:
when n is divided by 5, the remainder is 1 means : n=5X+1
when n is divided by 7, the remainder is 3 means : n=7y+3

here we exploit the "35" i.e., 5 and 7 are not given in vain: their product is 35;
by combining the above equations, we can solve the problem easily
for this(to combine), we put n on one side of the equation, on the other side of the equation, we initially write the product of 5 and 7: 35 => n=35T +... and further we insert 1 into x and y, finding the results of them, respectively: 5*1+1=6 and 7*1+3=10, and then we add the results 6 and 10. 16 goes to the end of the equation: n=35T+16 then what we should add to the right of the equation so that it becomes multiple of 35. 19 it is! k=19.

When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?

(A) 3
(B) 4
(C) 12
(D) 32
(E) 35

When positive integer n is divided by 5, the remainder is 1.
The smallest possible value of n that satisfies the statement above is the given remainder of 1.
To determine the other possible values of n, just keep adding multiples of the divisor 5:
1,6,11,16,21,26,31...

When positive integer n is divided by 7, the remainder is 3.
The smallest possible value of n that satisfies the statement above is the given remainder of 3.
To determine the other possible values of n, just keep adding multiples of the divisor 7:
3,10,17,24,31...

The smallest value included in both lists is n=31.

Now examine the answer choices.
When the correct answer choice is added to n=31, the sum will be a multiple of 35.
We can quickly see that the smallest possible value that will work is k=4 in answer choice B:
n+k = 31+4 = 35.

The correct answer is B.