n and k are positive integers, and when n is divided by k²

Target question: What is the value of n + k?

Given: when n is divided by k², the quotient is 5 and the remainder is 5k
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3
So, we can take the given information and write: n = 5k² + 5k

Statement 1: n = 550
Since we already know that n = 5k² + 5k, we can write: 550 = 5k² + 5k
Write as: 5k² + 5k - 550 = 0
Factor to get: 5(k² + k - 110) = 0
Factor again: 5(k + 11)(k - 10) = 0
This means that k = -11 or k = 10
We're told that k is POSITIVE, so it must be the case that k = 10
We can now conclude that n + k = 550 + 10 = 560
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: k = 10
Since we already know that n = 5k² + 5k, we can write: n = 5(10)² + 5(10) = 500 + 50 = 550
We can now conclude that n + k = 550 + 10 = 560
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer:

Cheers,
Brent