Question 3

When $$a=2^{2014}$$ and $$b=2^{2015}$$, prove that $$\left\{\frac{\frac{(a+b)^{2}+(a-b)^{2}}{b-a}-(a+b)}{\frac{1}{b-a}- \frac{1}{a+b}} \right\}\div \left\{\frac{(a+b)^{3}+(b-a)^{3}}{(a+b)^{2}-(a-b)^{2}} \right\}$$ is divisible by 3.

Correct Answer: 3a

Solution

Since $$b=2a$$, substituting throughout and simplifying each bracket step by step reduces the whole expression to $$3a$$. Since a is a positive integer (a power of 2), the expression $$3a$$ is clearly divisible by 3.

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When $$a=2^{2014}$$ and $$b=2^{2015}$$, prove that $$\left\{\frac{\frac{(a+b)^{2}+(a-b)^{2}}{b-a}-(a+b)}{\frac{1}{b-a}- \frac{1}{a+b}} \right\}\div \left\{\frac{(a+b)^{3}+(b-a)^{3}}{(a+b)^{2}-(a-b)^{2}} \right\}$$ is divisible by 3.

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