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
For many test-takers, the best approach will be to make a list of values for n.
When positive integer n is divided by the divisor D, the remainder is R.
Given the information above, the smallest possible value of n will be the remainder R.
To determine the other possible values of n, just keep adding multiples of the divisor D.
When positive integer n is divided by 5, the remainder is 1.
Smallest n = 1.
Now add multiples of 5:
1,6,11,16,21,26,31...
When positive integer n is divided by 7, the remainder is 3.
Smallest n = 3.
Now add multiples of 7:
3,10,17,24,31...
The smallest value included in both lists is n=31.
Now we can plug in the answers, which represent the smallest possible value of k.
When the correct answer is added to n=31, the sum will be a multiple of 35.
Since we need the smallest possible value of k, we should start with the smallest answer choice.
Answer choice A: k=3
n+k = 31+3 = 34. Not a multiple of 35.
Eliminate A.
Answer choice B: k=4
n+k = 31+4 = 35. Success!
The correct answer is
B.
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