Question Type 5: Using induction for proving divisibility Exercises

Prove by induction that 5 divides $3^{2n+1} + 2^{2n+1}$ for all integers $n\ge1$.

Prove by induction that $n^3 - n$ is divisible by $6$ for all integers $n\ge1$.

Prove by induction that 6 divides $7^n - 1$ for all integers $n\ge1$.

Prove by induction that 7 divides $6^{2n} - 1$ for all integers $n\ge1$.

Prove by induction that 13 divides $3^{12n} - 1$ for all integers $n\ge1$.

Prove by induction that 17 divides $2^{8n} - 1$ for all integers $n\ge1$.

Prove by induction that 7 divides $8^n - 1$ for all integers $n\ge1$.

Prove by induction that 3 divides $5^{2n} + 2^{2n+1}$ for all integers $n\ge1$.

Prove by induction that 31 divides $2^{5n} - 1$ for all integers $n\ge1$.

Prove by mathematical induction that $9$ divides $10^n - 1$ for all integers $n \ge 1$.

Prove by induction that 15 divides $2^{4n} - 1$ for all integers $n\ge1$.

Prove by induction that 13 divides $14^n - 1$ for all integers $n\ge1$.

Prove by induction that 3 divides $4^n - 1$ for all integers $n\ge1$.