Number Properties Question

[cssears]

What is the most efficient way to solve a question like this.

when positive interger N is divided by 5, the remainder is 1. When N is divided by 7, the remainder is 3. What is the smallest possible interger K such that K + N is a muliple of 35?

3
4
12
32
35

[akhp77]

N = 5a + 1
N = 7b + 3 = (7b + 2) + 1

LCM of 5 and 7 is 35

7b + 2 should be multiple of 5 at minimum value of b = 4
If we put b=4 in above equation, it will give the remainder (7*4 + 3 = 31) when N divided by 35.

So, we get
N = 35c + 31
N+4 = 35c + 31 + 4

K = 4 Answer

[jnelson0612]

Here's a quick and dirty way to do this problem. It's not elegant but it works:

When N is divided by 5, remainder is 1.
So N could be 6, 11, 16, 21, 26, 31, 36, 41, etc.

When N is divided by 7, the remainder is 3.
So N could be 10, 17, 24, 31, 38, etc.

What's your first common N from both lists? It is 31.

31 plus K is a multiple of 35. Well, let's make 31 + K = 35.

Thus, K = 4.