Question

If a and b distinct integers, prove that a-b is a factor of $$a^{n}-b^{n}$$, whenever n is a positive integer.

Answer 1

Hint: In this question it is given that if a and b distinct integers, then we have to prove that a-b is a factor of $$a^{n}-b^{n}$$, whenever n is a positive integer. So to prove this we need to know the binomial expansion, which is-
$$\left( a+b\right)^{n} =\ ^{n} C_{0}\ a^{n}+\ ^{n} C_{1}\ a^{n-1}b+\ ^{n} C_{2}\ a^{n-2}b^{2}+\cdots +\ ^{n} C_{n}\ b^{n}$$......(1)
And after that if we are able to show that $$a^{n}-b^{n}=\left( a-b\right) \cdot k$$ (where ‘k’ is a natural number) then we can easily say that a-b is a factor of $$a^{n}-b^{n}$$.
Complete step-by-step solution:
So in order to find the solution we can write $a^{n}$ as,
$$a^{n}=\left( a\right)^{n} $$
$$\Rightarrow a^{n}=\left( a-b+b\right)^{n} $$
$$\Rightarrow a^{n}=\left\{ b+\left( a-b\right) \right\}^{n} $$
Now applying the formula (1) by taking a=a and b=a-b, we get,
$$\Rightarrow a^{n}=\ ^{n} C_{0}\ b^{n}+\ ^{n} C_{1}\ b^{n-1}\left( a-b\right) +\ ^{n} C_{2}\ b^{n-2}\left( a-b\right)^{2} +\cdots \ +\ ^{n} C_{n}\left( a-b\right)^{n} $$
$$\Rightarrow a^{n}=b^{n}+\ ^{n} C_{1}\ b^{n-1}\left( a-b\right) +\ ^{n} C_{2}\ b^{n-2}\left( a-b\right)^{2} +\cdots \ +\ ^{n} C_{n}\left( a-b\right)^{n} $$ [since, $${}^{n}C_{0}=1$$]
$$\Rightarrow a^{n}-b^{n}=\ ^{n} C_{1}\ b^{n-1}\left( a-b\right) +\ ^{n} C_{2}\ b^{n-2}\left( a-b\right)^{2} +\cdots \ +\ ^{n} C_{n}\left( a-b\right)^{n} $$
$$\Rightarrow a^{n}-b^{n}=\ ^{n} C_{1}\ b^{n-1}\left( a-b\right) +\ ^{n} C_{2}\ b^{n-2}\left( a-b\right) \left( a-b\right) +\cdots \ +\ ^{n} C_{n}\left( a-b\right) \left( a-b\right)^{n-1} $$
$$\Rightarrow a^{n}-b^{n}=\left( a-b\right) [\ ^{n} C_{1}\ b^{n-1}+\ ^{n} C_{2}\ b^{n-2}\left( a-b\right) +\cdots \ +\ ^{n} C_{n}\left( a-b\right)^{n-1} ]$$
$$\Rightarrow a^{n}-b^{n}=\left( a-b\right) \cdot k$$
Where, $$k=[\ ^{n} C_{1}\ b^{n-1}+\ ^{n} C_{2}\ b^{n-2}\left( a-b\right) +\cdots \ +\ ^{n} C_{n}\left( a-b\right)^{n-1} ]$$ is a real numbers.
Therefore, we can say that a-b is a factor of $$a^{n}-b^{n}$$.
Hence proved.
Note: You can also solve this problem by using the Mathematical induction, i.e, firstly, consider an initial value for which the statement is true. It is to be shown that the statement is true for n = initial value=1.
Secondly, assume the statement is true for any value of n = k. Then prove the statement is true for n = k+1. We actually break n = k+1 into two parts, one part is n = k (which is already proved) and try to prove the other part.