**Answer:**

a) **4(9****+2)**

**b)6(7**** +1)**

**Step-by-step explanation:**

a) Prove that for any positive integer n, 4 evenly divides 32n-1

checking whether the statement is correct or not

∴ n = 1;

=

=

= 9 - 1

= 8

hence it is divisible by 4

Let the statement is for n = k

∴ = 4(equation 1)

= 4

= 4 (equation 1)

Now, we have to proof the statement is true for n = k+1

=

= ()

Adding & Subtracting 8

=

=

taking common 9

= 9( -1)+8

= 9 (4) +8 (from equation 1)

= 36 + 8

= **4(9****+2)**

if (9+2) = p

then = 4p

Since = 4p evenly divisible by 4

therefore given statement is true

b)Prove that for any positive integer n, 6 evenly divides

checking whether the statement is correct or not

∴ n = 1;

7 - 1

6

6 is divisible by 6

hence the given statement is true for n = 1

let it also true for n = k

(equation 2)

Now we have to proof the statement is true for n = k+1

Adding & Subtracting 6

7(6 )+6 ( from equation 2)

= 42 + 6

**= 6(7**** +1)**

if 6(7 +1) = p

then = 6p

Since = 6p evenly divisible by 6

therefore given statement is true