If a and b are distinct integers, prove that a - b is a factor of \(a^n - b^n\) , whenever n is a positive integer.
[Hint: write\( a ^n = (a - b + b)^n\) and expand]
To prove:
If a and b are distinct integers, then (a − b) is a factor of an − bn, where n is a positive integer.
Step 1: Rewrite an
a = (a − b) + b
∴ an = [(a − b) + b]n
Step 2: Expand using Binomial Theorem
[(a − b) + b]n = nC0(a − b)n + nC1(a − b)n−1b + nC2(a − b)n−2b2 + … + nCn−1(a − b)bn−1 + bn
Step 3: Subtract bn
an − bn =
(a − b)[ (a − b)n−1 + nC1(a − b)n−2b + nC2(a − b)n−3b2 + … + nCn−1bn−1 ]
Step 4: Factorization
Thus, an − bn is expressed as (a − b) multiplied by an integer expression.
Conclusion:
(a − b) is a factor of an − bn for all positive integers n.