Proposition IX.31 — If an odd number be prime to any number, it will also be pri…

Proof

g cd (a, n) = 1

for

a

odd implies

g cd (a, 2 n) = 1

(since

g cd (a, 2) = 1

); apply VII.24.

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Depends on (2)

VII.24Proposition VII.24If two numbers be prime to any number, their product also will be prime to the same.
VII.7Definition VII.7An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.

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