Determine whether the statement is true or false. If the statement is true, eval

Question

Determine whether the statement is true or false. If the statement is true, evaluate the proof. Is the proof correct? If not, which steps in the proof are wrong, and what would you have to do to fix it? If the statement is false find a counter example.

Theorem: Let a, b, c be integers. If a|b and a|c, then a|(b+2c).

Proof: If a|b and a|2c, then there is an integer q such that aq=b and a(2q)=2c. Then we have

b+2c = aq+2aq = 3aq = a(3q)

Since 3 and q are both integers, then 3q is also an integer, and a|(b+2c) by definition.

Explanation / Answer

The statement is true.

The proof is incorrect.

There is an integer q such that, aq = b , but a(2q) is not equal to 2c in general!

To coorect it, you should do the following steps:

a|2c , then there exists another integer k, such that:

ak = 2c

Then we have:

b+2c = aq + ak = a(q+k)

Since q+k is the sum of two integers, it is equal to an integer, and because b+2c is the multiplication of a by an integer, a|(b+2c) by definition.

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