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Prove Euclid's Lemma. (Hint: Use Prop 12.8)

Prove Euclid's Lemma. (Hint: Use Prop 12.8)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Proposition 12.8. If gcd(n, a) = = 1 and n|(a b) then n|b. Proof. Suppose that gcd(n, a) = 1 and n divides a b. By Bézout's Lemma, nx+ay = 1 for some x, y = Z. Multiplying both sides by b we get nxbaby b. Since n divides both terms on the left, it divides b as well. The following statement is known as Euclid's lemma (or Euclid's first theorem). It appears as Proposition 30 in Book VII of Euclid's Elements, written c. 300 BC.
Transcribed Image Text:Proposition 12.8. If gcd(n, a) = = 1 and n|(a b) then n|b. Proof. Suppose that gcd(n, a) = 1 and n divides a b. By Bézout's Lemma, nx+ay = 1 for some x, y = Z. Multiplying both sides by b we get nxbaby b. Since n divides both terms on the left, it divides b as well. The following statement is known as Euclid's lemma (or Euclid's first theorem). It appears as Proposition 30 in Book VII of Euclid's Elements, written c. 300 BC.
Prove Euclid's Lemma. (Hint: Use Prop 12.8)
Transcribed Image Text:Prove Euclid's Lemma. (Hint: Use Prop 12.8)
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