Divisibility Properties of Various Products of Integers
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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).
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Solution Summary
We prove certain divisibility properties of various integer products using modular arithmetic.
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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 ...
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