Algebra-15 (CAT-2002)

For all integers n>0, 7⁶ⁿ-6⁶ⁿ is divisible by:

(1)13  (2)128  (3)549  (4) None of these

For Answer Click on "Read more" below:

Solution:

7⁶ⁿ-6⁶ⁿ = (7³ⁿ+6³ⁿ) (7³ⁿ-6³ⁿ)

= (7ⁿ+6ⁿ) (7²ⁿ+6²ⁿ-7ⁿ6ⁿ) (7ⁿ-6ⁿ) (7²ⁿ+6²ⁿ+7ⁿ6ⁿ)

= (7ⁿ+6ⁿ) (7ⁿ-6ⁿ) (7²ⁿ+6²ⁿ-7ⁿ6ⁿ) (7²ⁿ+6²ⁿ+7ⁿ6ⁿ)

For odd n, (7ⁿ+6ⁿ) is divisible by 7+6 ie., 13

For even n, (7ⁿ-6ⁿ) is divisible by 7+6 ie., 13

Hence the given number is divisible by 13 for all n

Answer(1)