Use any mathematical induction to prove that for any positive integer n, 6^n-1 is divisible by 5. (also in Description below)

Question

Use any mathematical induction to prove that for any positive integer n, 6ⁿ-1 is divisible by 5.

Answer 1

n=1 --> 6^1-1 = 5

n=2 --> 6^2-1 = 36-1 = 35 is divisible by 5

suppose 6^k-1 is divisible by 5

6^(k+1)-1 = 6^k*6-1 = 6^k*6 -1 +6 - 6

= 6^k*6 - 6 + 6 - 1

= (6^k*6 - 6) + (6-1)

= 6( 6^k -1) + 5

by induction hypothesis 6^k-1 is divisible by 5;

so 6^k-1 = 5*T for some integer T

substituting 6 ( 5T) + 5 = 5 ( 6T+1)

again by closure property of multiplication and addition

of integers 6T+1 is an integer, so the result is

even divisible by 5