Question

A whole number is input into the machine The output is then used as a new input to produce a second output. Rosa says "the last digit of the second output must be a 5." Is Rosa correct? Explain your answer

Answer 1

No Rosa, is not correct. See and explanation below

Explanation:

Let `e` be an even number and put it into the first machine.

`e + 3` will be an odd number.

`e + 3 = e + 2 + 1`

An even number plus `2` will be another even number.

Then, an even number plus `1` will be an odd number.

Therefore `e + 3 =` and odd number#

Let `d` be an even number and put it into the first machine.

`d + 3` will be an even number.

`d + 3 = d + 2 + 1`

An odd number plus `2` will be another odd number.

Then, an odd number plus `1` will be an even number.

Therefore `d + 3 =` an even number#

Therefore Machine 1 can output an even or an odd number

Let `d` be an odd number and put it into the second machine.

`d xx 5 =` a number ending in `5`

An odd number times `5` always ends in `5`

`5 xx 1 = 5`

`5 xx 3 = 15`

`5 xx 5 = 25`

`5 xx 7 = 35`

`5 xx 9 = 45`

Let `e` be an odd number and put it into the second machine.

`e xx 5 =` a number ending in `0`

Because `e` is even it can be written as `2 xx i` where `i` is an integer:

Then:

`5 xx e = 5 xx (2 xx i) = 10 xx i = i0`

Therefore Machine 2 can output a number ending in `0` or `5`