Factor reminder math question.

Question

When a negative integer x is divided by -5, the remainder is 1. When x is divided by 8, the reminder is -4. What is the largest value of the of the negative integer y such that x multiplied by y is divisible by 40.

Answer 1

We are given:

(1) x < 0

(2) There exists an integer a so that x = -5a + 1

(3) There exists an integer b so that x = 8b - 4

By observation, x = -4 is the largest solution to (1), (2), (3).

Since 5 and 8 are relatively prime and 5×8 = 40, x must be of the form

x = 40c - 4

for some integer c ≤ 0.

For any y,

xy = 40cy - 4y

The requirement that xy be divisible by 40 is then equivalent to the requirement

that 4y be divisible by 40.

The largest negative integer satisfying that is y = -10.