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

Answer:

0

Step-by-step explanation:

We are given the following:

$\frac{x}{5}=q_1+\frac{4}{5}$      (equ. 1)

$\frac{y}{5}=q_2+\frac{1}{5}$      (equ. 2)

We are asked:

$\frac{x+y}{5}=q_3+\frac{r}{5}$  (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.