SOLUTION: 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

Algebra ->  Divisibility and Prime Numbers  -> Lessons -> SOLUTION: 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      Log On


   



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

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
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

Let all letters represent non-negative integers with x%3E0

If y divided by x gives quotient q and remainder r, then y+=+xq+%2B+r

Conversely, if y+=+xq+%2B+r and r%3Cx, then y divided by x
gives quotient q and remainder r. 

Suppose a divided by 4 gives quotient q%5B1%5D and 
suppose b divided by 4 gives quotient q%5B2%5D, then

system%28a=4q%5B1%5D%2B2%2Cb=4q%5B2%5D%2B3%29

Adding the two equations,

a%2Bb=4q%5B1%5D%2B4q%5B2%5D%2B5

Write 5 as 4%2B1

a%2Bb=4q%5B1%5D%2B4q%5B2%5D%2B4%2B1

a%2Bb=4%28q%5B1%5D%2Bq%5B2%5D%2B1%29%2B1

By the converse above, a%2Bb divided by 4 gives quotient
q%5B1%5D%2Bq%5B2%5D%2B1 and remainder 1.

That's choice b.

Edwin