1. ## Division

This is just a confirmation of the work I have done:

Indicate if the following are true or false. If false, give a counterexample:

1.) If b|(a+b), then d|a or d|b

False: 3|(7+2), but 3 does NOT | 7 or 3 does NOT | 2

2.) If ab|c, a =! 0 and b =! 0, then a|c and b|c

True

3.) If d|a and d|b, then d|(ax + by) for all integers x and y

True

4.) If d|a^2, then d|a

True

2. 4.) false
Take a = 6, then a^2 = 12. 4|12, but not 6.

3. Originally Posted by terr13
4.) false
Take a = 6, then a^2 = 12. 4|12, but not 6.
Uh, 6^2 = 36, not 12.

4. Wow, I'm bad at math. Take a=4, then a^2= 16. 8|16, but 8 does not divide 4.

5. 1.) If b|(a+b), then d|a or d|b

False: 3|(7+2), but 3 does NOT | 7 or 3 does NOT | 2

2.) If ab|c, a =! 0 and b =! 0, then a|c and b|c

True

3.) If d|a and d|b, then d|(ax + by) for all integers x and y

True

4.) If d|a^2, then d|a

True

Look at terr13's example

6. Originally Posted by DiscreteW
4.) If d|a^2, then d|a

True
this is not true unless d is a prime number

7. That is essentially the definition of a prime element, if a is prime and a|bc, then a|b or a|c.