Patulong po ako nito, below is the example of this.

Example: Prove that 230 -1 is divisible by 7 a Verification 1 = 1 3 (1 ) 1 = 4 230- 1 = 2 - 1 = 7 3( 4 ) 237- 1 - 2 - 1 7/7 7/ 4095 b. Induction hypothesis 7123k - 1 c. Proof of the Induction 7/73 ( k+1 ) 3 (K+ 1 ) 3k+03 2 - 1 = 2 3k 3 - 1 = 2 2 - 1 = 8025k - 1 = 8.234- 8+7 = 8 (25K- 1 ) +7 25k is divisible by 7 based on our Induction hypothesis. Hence, 8 (23k ,) is divisible by 7 Therefore , 8 ( 23 k ,3 ( k +1) lit7 is divisible by 7. This means - I is divisible by. d . conclusion 23- 1 is divisible by 7.Theory of Numbers Instructions: Prove the following using mathematical induction. 1.) 3-1 is divisible by 8 for all natural number n. a . verification : b. Induction Hypothesis : C. Proof of the Induction . d Conclusion : 2. ) 10"+ 3 /40+2)+5 is divisible by a for all positive integral values of n. a. verification: b. Induction Hypothesis: C. proof of the induction: d. conclusion : 3. 13 +6n2- 4n +3 is divisible by 3 for any natural number n. a . Verification : b. Induction Hypothesis C. Proof of the Induction : d. conclusion : 4. 72"- 48n-1 is divisible by 2304 for every natural integer n. a. Verification: b. Induction Hypothesis : C. Proof of the Hypothesis : d- conclusion : 5 . azn - 1 +b 27 - 1 is divisible by atb, for n is a positive integer 9. Verification : b. Induction Hypothesis: c. proof of the Induction : d conclusion