Definition 9 (Euclid VII, Def. 6). An even number is that which is divisible into two equal parts.
Even and Odd
Definitions 9–10 · Theorems 23–29
Definition 10 (Euclid VII, Def. 7). An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.
Theorem 23 (Euclid IX.21). If as many even numbers as we please be added together, the whole is even.
Proof: Each has a half part, so the whole has a half part.
Theorem 24 (Euclid IX.22). If as many odd numbers as we please be added together, and their multitude be even, the whole will be even.
Proof: Subtract a unit from each. Each remainder is even, so their sum is even (Theorem 23). The multitude of the units is even. So the whole is even (Theorem 23).
Theorem 25 (Euclid IX.23). If as many odd numbers as we please be added together, and their multitude be odd, the whole will also be odd.
Proof: Let them be AB, BC, CD. Subtract the unit DE from CD. CE is even, and CA is even (Theorem 24), so AE is even (Theorem 23). With the unit, AD is odd.
Theorem 26 (Euclid IX.24). If from an even number an even number be subtracted, the remainder will be even.
Proof: The whole has a half part and the number subtracted has a half part, so the remainder has a half part.
Theorem 27 (Euclid IX.25). If from an even number an odd number be subtracted, the remainder will be odd.
Proof: From the even AB let the odd BC be subtracted. Subtract the unit CD from BC. DB is even, so the remainder AD is even (Theorem 26). CD is a unit, so CA is odd.
Theorem 28 (Euclid IX.26). If from an odd number an odd number be subtracted, the remainder will be even.
Proof: From the odd AB let the odd BC be subtracted. Subtract the unit BD from each. AD and CD are even, so the remainder CA is even (Theorem 26).
Theorem 29 (Euclid IX.27). If from an odd number an even number be subtracted, the remainder will be odd.
Proof: From the odd AB let the even BC be subtracted. Subtract the unit AD. DB is even, so the remainder CD is even (Theorem 26), and CA is odd.