SOLUTION: 4) A student claims �If d does not divide a and d does not divide b, then d does not divide (a+b). How do you respond? Hint look in chapter 4.

Question 176318: 4) A student claims �If d does not divide a and d does not divide b, then d does not divide (a+b). How do you respond? Hint look in chapter 4.
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
The claim is false. Here's a counter-example to show why:

Let d=10, a=3, and b=7 (you can pick any random numbers as long as d does not divide either a or b, but it divides the sum)

It is true that d does NOT divide a (since 10 does NOT divide 3) and d does NOT divide b (since 10 does NOT divide 7).

However, d does divide a+b since 10 divides (3+7) or 10. In other words,

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The modulo (x mod y) function provides the remainder when one integer (x) is divided by another (y). So if d does not divide a, then a mod d is non-zero. Also, if d does not divide b, then b mod d is non-zero. However, it is possible that (a mod d) + (b mod d) = kd where k is a non-zero positive integer.

Further (a + b) mod d = (a mod d) + (b mod d) so (a + b) is divisible by d whenever (a mod d) + (b mod d) = kd.
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