Question 117

If a and b are two odd positive integers, by which of the following integers is $$(a^4-b^4)$$ always divisible ?

Solution

$$(a^4-b^4) = (a^2+b^2)\times(a^2-b^2)$$
= $$(a^2+b^2)\times(a+b)\times(a-b)$$

Let a = 5 , b= 3
54 - 34 = (52 + 32)(5 + 3)(5 - 3) = 34 x 8 x 2 which is divisible by 8
Let a = 7 and b = 5
74 - 54 = (72 + 52)(7 + 5)(7 - 5) = 74 x 12 x 2 which is also divisible by 8

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