The hints for (a) Use Proposition 5.5.16. (b): p must divide the left-handside of the multiplied equation (explain why). (c): Consider two cases (I) ais relatively prime to p; (II) a is not relatively prime to pPlease show step by step (a) Let p be a prime, and let a be an integer. Show that a is relatively primeto p if and only if there exist integers m and n such that pm + an = 1.(*Hint*)(b) Suppose p is prime, and suppose a is relatively prime to p. Suppose alsothat p divides ab. By multiplying the equation in part (a) by b, showthat p must divide b. (*Hint*)(c) Prove Euclid's Lemma: Let p be a prime number, and let a and b beintegers. If p divides ab, then either p divides a or p divides b. (*Hint*) Proposition 5.5.16. Given the Diophantine equation an + bm = c, wherea, b, c are integers. Then the equation has integer solutions for n and m ifand only if c is a multiple of the gcd of a and b.PROOF. Since this is an "if and only if" proof, we need to prove it bothways. We'll do “only if" here, and leave the other way as an exercise.Since we're doing the "only if" part, we assume that an + bm = c issolvable. We'll represent the gcd of a and b by the letter d. Since gcd(a, b)divides both a and b, we may write aa', b'. By basic algebra, we have an + bm =this back in the original Diophantine equation, we get:da' and b = db' for some integersd(a'n + b'm). If we substituted(a'n + b'm) =It follows that c is a multiple of, d, which is the gcd of a and b.

Answered: Image /qna-images/answer/edba3f7a-fff5-4e66-9ad3-814b9eb08c05.jpg

(a) Let p be a prime, and let a be an integer. Show that a is relatively prime to p if and only if there exist integers m and n such that pm + an = 1. (*Hint*) (b) Suppose p is prime, and suppose a is relatively prime to p. Suppose also that p divides ab. By multiplying the equation in part (a) by b, show that p must divide b. (*Hint*) (c) Prove Euclid's Lemma: Let p be a prime number, and let a and b be integers. If p divides ab, then either p divides a or p divides b. (*Hint*)

(a) Let p be a prime, and let a be an integer. Show that a is relatively prime to p if and only if there exist integers m and n such that pm + an = 1. (Hint) (b) Suppose p is prime, and suppose a is relatively prime to p. Suppose also that p divides ab. By multiplying the equation in part (a) by b, show that p must divide b. (Hint) (c) Prove Euclid's Lemma: Let p be a prime number, and let a and b be integers. If p divides ab, then either p divides a or p divides b. (Hint)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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