For every positive integer n, prove that 7 power n minus 3 power n is

1.	For every positive integer n, prove that 7 power n minus 3 power n is divisible by 4
Answer» Step 1,For n =17^1-3^1=4.which is divisible by 4.Step 2,Given statement is true for n=1,It must be true for n=k, 7^k-3^k=(M) 4 [As it is divisible by 4, let =(M) 4 where M is a positive integer.]...... (1)Step 3,For n=k+1,7^(k+1)-3^(k+1)7^k×7^1-3^k×3^1Using equation (1) {(M) 4+3^k} ×7^1-3^k×3^1(M) 4×7+3^k×7-3^k×37(M)4+3^k(7-3)7(M)4+3^k×47(M)4+3^k×(M)47 is multiplied with number which is divisible by 4 and added to positive number which is also divisible by 4.Therefore the resultant is also a divisible by 4.

Discussion