Kitap 9
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For from the even number AB let the odd number BC be subtracted; I say that the remainder CA is odd.
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For let the unit CD be subtracted from BC; therefore DB is even. [VII. Def. 7]
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But AB is also even; therefore the remainder AD is also even. [IX. 24]
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And CD is an unit; therefore CA is odd. [VII. Def. 7] Q. E. D.
PROPOSITION 26.
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If from an odd number an odd number be subtracted, the remainder will be even.
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For from the odd number AB let the odd number BC be subtracted; I say that the remainder CA is even.
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For, since AB is odd, let the unit BD be subtracted; therefore the remainder AD is even. [VII. Def. 7]
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For the same reason CD is also even; [VII. Def. 7] so that the remainder CA is also even. [IX. 24] Q. E. D.
PROPOSITION 27.
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If from an odd number an even number be subtracted, the remainder will be odd.
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For from the odd number AB let the even number BC be subtracted; I say that the remainder CA is odd.
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Let the unit AD be subtracted; therefore DB is even. [VII. Def. 7]
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But BC is also even; therefore the remainder CD is even. [ IX. 24 ]
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Therefore CA is odd. [VII. Def. 7] Q. E. D.
PROPOSITION 28.
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If an odd number by multiplying an even number make some number, the product will be even.
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For let the odd number A by multiplying the even number B make C; I say that C is even.
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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]
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And B is even; therefore C is made up of even numbers.
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But, if as many even numbers as we please be added together, the whole is even. [IX. 21]
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Therefore C is even. Q. E. D.
PROPOSITION 29.
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If an odd number by multiplying an odd number make some number, the product will be odd.
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For let the odd number A by multiplying the odd number B make C; I say that C is odd.
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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]
