# Math Help - Questions Involving Divisiblity

1. ## Questions Involving Divisiblity

Hi,

Could someone help me solve this, please.

Show by Induction that for n >= 1:
(3^(3n)) + (2^(n+2)) is divisible by 5

Thanks.

2. Originally Posted by justmaths
Hi,

Could someone help me solve this, please.

Show by Induction that for n >= 1:
(3^(3n)) + (2^(n+2)) is divisible by 5

Thanks.
True for n = 1.

Assume true for n = k.

Show that under the assumption it's true for n = k + 1:

$3^{3(k+1)} + 2^{(k+1)+2} = 3^{3k} \cdot 3^3 + 2 \cdot 2^{k+2} = 27 \cdot 3^{3k} + 2 \cdot 2^{k+2}$

$= 25 \cdot 3^{3k} + 2 \cdot 3^{3k} + 2 \cdot 2^{k+2} = 25 \cdot 3^{3k} + 2 \left( 3^{3k} + 2^{k+2} \right)$

and it should be clear how to finish things off.

3. Oh I see. Then we let the last bracket equal a number divisible by 5.

Thanks.