Question

Prove the divisibility of the following numbers: 45^10·5^40 by 25^20

Answer 1

For this case we have the following expressions:

`45^(10) * 5^ {40}` and
`25^(20)`

Let:

`45^(10) * 5^ {40}` (Integer m)

`25^(20)` (Integer n)

By definition, an integer m is divisible by an integer n if the remainder of the division is 0. That is, there is an integer p such that:
`m = n * p`

Rewriting the expression we have:

`25^(20) = 5^(2 * (20)) = 5^( 40)`

So:

`\frac {45^(10) * 5^( 40)} {5^(40)} =`

We cancel similar terms:

`45^(10)`

We check:

`45^(10) * 5^( 40) = 5^( 40) * 45^( 10)`

Answer:

If they are divisible