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.