Deduction with Integers

A direct proof begins with the assumptions and ends with the claimed property. Translate words into algebra: an even integer is \(2k\), an odd integer is \(2k+1\), and a multiple of \(d\) is \(dk\), where \(k\) is an integer.

When two integers are independent, use different letters for them. A sum or product of integers is an integer, so displaying a factor of \(d\) proves divisibility by \(d\).

Example

For even integers \(2j\) and \(2k\), their sum is \(2j+2k=2(j+k)\). Since \(j+k\) is an integer, the sum is even.

Write why the final expression has the required property. Finite numerical checks support error detection; the general algebra provides the proof.

Notation used in the examples

For integers \(d\ne0\) and \(a\), the notation \(d\mid a\) means that \(a=dk\) for some integer \(k\): in words, \(d\) divides \(a\).

The notation \(n^2\equiv1\pmod 8\) means that \(n^2\) leaves remainder \(1\) when divided by \(8\), or equivalently \(n^2=8k+1\) for an integer \(k\). Here this is only shorthand for the remainder statement; use ordinary integer algebra in the proof.