Proofs:Direct

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The direct proof is relatively simple — by logically applying previous knowledge, we directly prove what is required.

Example 1

Prove that the sum of any two even integers and is even.

Solution 1

We know that since and are even, they must have 2 as a factor. Then, we can write the following:

Let , , for some integers

Then:

by the distributive property of integers

The number clearly has 2 as a factor, which implies it is even. Therefore, is even.

Example 2

Prove the following statement for non-zero integers :

If divides and divides , then divides .

Solution 2

If an integer divides an integer , then we can write , for some non-zero integer . So let's say that and , for some non-zero integers and . Then:

by the associative property of integer multiplication.

But since and are integers, their product must also be an integer. Therefore, is the product of some integer multiplied by , so we get that divides .

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