Question

 · If P(n): "2 42n+1+33n+1 is divisible by 'lambda' for all n element of N" is true, then find the value of 'lambda'

Answer 1

Hint mod 11: 8⋅16n+3⋅27n≡11⋅16n≡0 mod 11: 8⋅16n+3⋅27n≡11⋅16n≡0 by 27≡16 27≡16

Remark If modular arithmetic is unfamiliar then you can rewrite is as follows

3(27n−16n)+11⋅16n

3(27n−16n)+11⋅16n

which is divisible by 1111 by 11=27−16∣27n−16n11=27−16∣27n−16n (provable by induction or by invoking the Factor Theorem a−b∣an−bn)a−b∣an−bn)

Induction is also (implicitly) used in the modular proof because it uses 27≡16⇒27n≡16n27≡16⇒27n≡16n and the proof of this Congruence Power Rule is by induction on nn.