Purchase Solution

# Divisibility Properties of Various Products of Integers

Not what you're looking for?

Prove that if x + y is even, then the product xy(x + y)(x - y) is divisible by 24, and that without this restriction, 4xy(x- y)(x + y) is divisible by 24. Consider that any integer is of the form 3k, 3k + 1, or 3k + 2 in showing that 3|xy(x + y)(x - y). Similarly, because any integer is of the form 8k, 8k + 1, ..., or 8k + 7, then 8|xy(x - y)(x + y).

##### Solution Summary

We prove certain divisibility properties of various integer products using modular arithmetic.

##### Solution Preview

If either x or y is divisible by 3, then clearly 3 divides xy(x + y)(x - y). On the other hand, suppose neither x nor y is divisible by 3. Then either x is congruent to y or -y modulo 3, in which case 3 divides either x - y or x + y respectively. Thus we see that in all cases we have 3|xy(x + y)(x - y).

First we consider the case in which x + y is even. Now if ...

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts