About Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, 4 evenly divides 32n-1. (b) Prove that for any positive integer n, 6 evenly divides 7n - 1. (c) Prove that for any positive integer n, 4 evenly divides 11n - 7n. (d) Prove that for any positive integer n, 7 evenly divides 9n - 2n.

(a)

(b)

(c)

(d)

Step-by-step explanation:

Solving (a): For integer , 4 divides

The proof is as follows:

For n = k, we have:

Multiply through by 4

Next, prove the statement is true for

Express -1 as - 9 + 8

Factorize:

Recall that:

So, we have:

Factorize

Since the above mathematical statement is true, then the given statement has been proved

Solving (b): For integer , 6 divides

The proof is as follows:

For n = k, we have:

Multiply through by 6

Next, prove the statement is true for

Express -1 as - 7 + 6

Factorize:

Recall that:

So, we have:

Factorize

Since the above mathematical statement is true, then the given statement has been proved

Solving (c): For integer , 4 divides

The proof is as follows:

For n = k, we have:

Multiply through by 4

Next, prove the statement is true for

Collect Like Terms

Recall that:

So, we have:

Factorize

Since the above mathematical statement is true, then the given statement has been proved

Solving (d): For integer , 7 divides

The proof is as follows:

For n = k, we have:

Multiply through by 7

Next, prove the statement is true for

Collect Like Terms

Recall that:

So, we have:

Factorize

Since the above mathematical statement is true, then the given statement has been proved