IB Mathematics Analysis and Approaches Question Type 5: Using Induction for Proving Divisibility Exercises

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

Prove by induction that $n^3-n$ is divisible by $6$ for all integers $n\ge 1$.

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

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

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

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

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

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

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

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 $n\ge 1$.

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

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