Question

Help me please on my RSM homework!

The questions are:

(^ = to the power)

1. 36^5 - 6^9 is divisible by 30

2. 5^18 - 25^8 is divisible by 120

Thanks very much!

Answer 1

1.
`36^5-6^9=(6^2)^5-6^9=6^(10)-6^9=6^9(6-1)=6^9\cdot5`

Then with
`30=6\cdot5`, it's easy to see the above is divisible by 30.

2.
`5^(18)-25^8=5^(18)-(5^2)^8=5^(18)-5^(16)=5^(16)(5^2-1)=5^(16)\cdot24`

`120=5\cdot24` and the claim follows.