PROPOSITION 27.
If from an odd number an even number be subtracted, the remainder will be odd.
For from the odd number
AB let the even number
BC be subtracted; I say that the remainder
CA is odd.
Let the unit
AD be subtracted; therefore
DB is even. [
VII. Def. 7]
But
BC is also even; therefore the remainder
CD is even. [
IX. 24
]
Therefore
CA is odd. [
VII. Def. 7] Q. E. D.
PROPOSITION 28.
If an odd number by multiplying an even number make some number, the product will be even.
For let the odd number
A by multiplying the even number
B make
C; I say that
C is even.
For, since
A by multiplying
B has made
C, therefore
C is made up of as many numbers equal to
B as there are units in
A. [
VII. Def. 15]
And
B is even; therefore
C is made up of even numbers.
But, if as many even numbers as we please be added together, the whole is even. [
IX. 21]
Therefore
C is even. Q. E. D.
PROPOSITION 29.
If an odd number by multiplying an odd number make some number, the product will be odd.
For let the odd number
A by multiplying the odd number
B make
C; I say that
C is odd.
For, since
A by multiplying
B has made
C,