Question 286764
If a divided by 4 leaves a remainder of 2 and b divided by 4 leaves a remainder of 3, then when a + b is divided by 4, the remainder is

A. 0 B. 1 C. 2 D. 3 E. 4

<pre><font size = 4 color = "indigo"><b>
Let all letters represent non-negative integers with {{{x>0}}}

If {{{y}}} divided by {{{x}}} gives quotient {{{q}}} and remainder {{{r}}}, then {{{y = xq + r}}}

Conversely, if {{{y = xq + r}}} and {{{r<x}}}, then {{{y}}} divided by {{{x}}}
gives quotient {{{q}}} and remainder {{{r}}}. 

Suppose {{{a}}} divided by {{{4}}} gives quotient {{{q[1]}}} and 
suppose {{{b}}} divided by {{{4}}} gives quotient {{{q[2]}}}, then

{{{system(a=4q[1]+2,b=4q[2]+3)}}}

Adding the two equations,

{{{a+b=4q[1]+4q[2]+5}}}

Write {{{5}}} as {{{4+1}}}

{{{a+b=4q[1]+4q[2]+4+1}}}

{{{a+b=4(q[1]+q[2]+1)+1}}}

By the converse above, {{{a+b}}} divided by {{{4}}} gives quotient
{{{q[1]+q[2]+1}}} and remainder {{{1}}}.

That's choice b.

Edwin</pre>