Question

Prove the divisibility of the following numbers:

1. 25^7+5^13 by 30
2. 7^6+7^5−7^4 by 11
3. 5^5−5^4+5^3 by 7

Answer 1

1. Make use of prime factorizations:

`25^7+5^(13)=(5^2)^7+5^(13)=5^(14)+5^(13)`

Pull out a common factor of
`5^(13)`:

`5^(14)+5^(13)=5^(13)(5+1)=5^(13)\cdot6`

Since
`30=5\cdot 6`, it follows that
`30\mid25^7+5^(13)`.

2. Pull out a common factor of
`7^4`:

`7^6+7^5-7^4=7^4(7^2+7-1)=7^4\cdot55=5\cdot7^4\cdot11`

3. Pull out a common factor of
`5^3`:

`5^5-5^4+5^3=5^3(5^2-5+1)=5^3\cdot21=3\cdot5^3\cdot7`