If as many odd numbers as we please be added together, and their multitude be even, the whole will be even.
For let as many odd numbers as we please, AB
, even in multitude, be added together; I say that the whole AE
For, since each of the numbers AB
is odd, if an unit be subtracted from each, each of the remainders will be even; [VII. Def. 7
] so that the sum of them will be even. [IX. 21
But the multitude of the units is also even.
Therefore the whole AE
is also even. [IX. 21
] Q. E. D.