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
There's a nice rule that says,
If, when N is divided by D, the remainder is R, then the possible values of N include: R, R+D, R+2D, R+3D,. . .
When n is divided by 5, the remainder is 1.
So, possible values of n are 1, 6, 11, 16, 21, 26,
31, 36, 41, 46, 51, 56, 61,
66, 71, 76, etc.
When n is divided by 7, the remainder is 3.
So, possible values of n are 3, 10, 17, 24,
31, 38, 45, 52, 59,
66, 73, etc.
So, we can see that n could equal 31, or 66, or an infinite number of other values.
Important: Since the Least Common Multiple of 7 and 5 is 35, we can conclude that if we list the possible values of n, each value will be 35 greater than the last value.
So, n could equal 31, 66, 101, 136, and so on.
Check the answer choices....
Answer choice A: If we add 3 to any of these possible n-values, the sum is NOT a multiple of 35.
ELIMINATE A
Answer choice B: if we take ANY of these possible n-values, and add 4, the sum will be a multiple of 35.
So, the smallest value of k is
4 such that k+n is a multiple of 35.
Answer =
B
Cheers,
Brent
