Task B-1.5. (20 points)

The difference of two odd numbers is divisible by 5. What is the last digit of the difference of the cubes of these numbers?
Solution. Let's denote these numbers as $a=2m-1, b=2n-1$. Their difference is

$$
a-b=2(m-n)
$$

(5 points)

Since the difference is divisible by 5 and by 2, it is divisible by 10, i.e., $a-b=10k$.

(5 points)

By factoring the difference of cubes $a^3-b^3=(a-b)(a^2+ab+b^2)$, we get $a^3-b^3=10k(a^2+ab+b^2)$, so $a^3-b^3$ is divisible by 10, meaning the difference of cubes ends in 0.

(10 points)

## COUNTY COMPETITION IN MATHEMATICS

## 2nd grade - high school - B category

## February 23, 2009.