Proof-focused number theory

Number theory studies integers through direct proof: since any even number is 2m and any other is 2n for some integers m and n, their sum 2m + 2n = 2(m+n) is also even, proving the claim for every case at once. The Euclidean algorithm finds the greatest common divisor by repeated division with remainder: for 48 and 18, 48 = 2(18)+12, then 18 = 1(12)+6, then 12 = 2(6)+0, so the greatest common divisor of 48 and 18 is 6.

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