Respuesta :
Answer:
0
Step-by-step explanation:
We are given the following:
[tex]\frac{x}{5}=q_1+\frac{4}{5}[/tex] (equ. 1)
[tex]\frac{y}{5}=q_2+\frac{1}{5}[/tex] (equ. 2)
We are asked:
[tex]\frac{x+y}{5}=q_3+\frac{r}{5}[/tex] (equ. 3) , what is [tex]r[/tex]?
The [tex]q_i[/tex]'s represent the quotients you get.
[tex]r[/tex] is the remainder of dividing [tex]x+y[/tex] by 5.
We know that [tex]r[/tex] is a number in {0,1,2,3,4}.
[tex]x=5q_1+4[/tex] (I got this by multiplying both sides of equ 1. by 5.)
[tex]y=5q_2+1[/tex] (I got this by multiplying both sides of equ 2. by 5.)
Let's add these equations together:
[tex]x+y=(5q_1+5q_2)+(4+1)[/tex]
Factoring the 5 out for the [tex]q_i[/tex]'s part and simplify 4+1 gives:
[tex]x+y=5(q_1+q_2)+5[/tex]
So [tex]5[/tex] can't be the remainder of dividing something by 5 but see that we can factor this right hand expression more as:
[tex]x+y=5(q_1+q_2+1)[/tex]
So [tex]q_3=q_1+q_2+1[/tex] 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.
