Reason Quantitatively Integers

Question

Which two integers have a difference of 24 and a quotient of -7?

Note: (I was thinking there wasn't any answer because I believe, there are no two numbers that divide together and equal -7. However I'm not sure.)

Answer 1

The two integers must satisfy

n - m = 24

n/m = -7

n/m = -7 ⇒ n = -7m

-7m - m = 24

-8m = 24

m = -3

n = 24 + m = 24 - 3 = 21

n = 21

Test the answer.

n - m = 21 - (-3) = 24 (good)

n/m = 21/-3 = -7 (good)

Try m/n = -7. Could this also give a solution?

m/n = -7 ⇒ m = -7n

n - (-7n) = 24

8n = 24

n = 3

m = -7n = -7(3) = -21

Test this answer too.

n - m = 3 - (-21) = 24 (good)

m/n = -21/-3 = -7 (good)

So you can see there are integer solutions.