Question

If x is divided by 5 the remainder is 4. If why is divided by 5 the remainder is 1. What is the remainder when x+y is divided by 5

Answer 1

0

Explanation:

We are given the following:

`(x)/(5)=q_1+(4)/(5)` (equ. 1)

`(y)/(5)=q_2+(1)/(5)` (equ. 2)

We are asked:

`(x+y)/(5)=q_3+(r)/(5)` (equ. 3) , what is
`r`?

The
`q_i`'s represent the quotients you get.

`r` is the remainder of dividing
`x+y` by 5.

We know that
`r` is a number in {0,1,2,3,4}.

`x=5q_1+4` (I got this by multiplying both sides of equ 1. by 5.)

`y=5q_2+1` (I got this by multiplying both sides of equ 2. by 5.)

Let's add these equations together:

`x+y=(5q_1+5q_2)+(4+1)`

Factoring the 5 out for the
`q_i`'s part and simplify 4+1 gives:

`x+y=5(q_1+q_2)+5`

So
`5` can't be the remainder of dividing something by 5 but see that we can factor this right hand expression more as:

`x+y=5(q_1+q_2+1)`

So
`q_3=q_1+q_2+1` while there is no remainder (the remainder is 0).

Let's do an example if you are not convinced at this point that the remainder will be 0.

So choose x=9 since 9/5 gives a remainder of 4.

And choose y=16 since 16/5 gives a remainder of 1.

x+y=9+16=25 and 25/5 gives a remainder of 0.