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4. Suppose that a₁,..., am are integers, not all 0, and let d = gcd(a₁,...,am). Show that dZ = a₁Z+a₂Z+...+amZ = {a₁u₁ +...+amum | uj € Z,1 ≤ j ≤ m}.

4. Suppose that a₁,..., am are integers, not all 0, and let d = gcd(a₁,...,am). Show that dZ = a₁Z+a₂Z+...+amZ = {a₁u₁ +...+amum | uj € Z,1 ≤ j ≤ m}.

Advanced Engineering Mathematics
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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[Number Theory] How do you solve question 4?
and the sum for the set of all multiples of k. 1. The Fibonacci numbers are defined by F₁ = 1, F2 = 1, F3 = F₁+F2 = 2, and, in general, for n ≥ 3, Fn = Fn-1 + Fn-2. Thus, the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, .. What is the sum F₁+F3+ + F2n+1? 2. Show that kZ = Prove your answers by induction on n. = { kn\n €Z} (i) 2 Fn Hint: For example in case (ii), write Fn Tn's are related. Here, for k EZ, we will use the notation F₂+F4 + + F2n? 3|n, (ii) 3 F 4|n, (iii) 4Fn 6 | n. So the sequence looks like Prove that - 3. Suppose that a and b are nonzero integers and let = lcm(a, b) be their least common multiple. Show that 3qn+rn with 0≤rn <3 and consider how the 4. Suppose that a₁,..., am are integers, not all 0, and let d = gcd(a₁,...,am). Show that dZ = a₁Z+ a2₂Z + ...+amZ = {a₁₁ + +amum | uj € Z, 1 ≤j≤m}. azn bz=lZ. 5. (i) Define a sequence of numbers Rn, n = 1,2,... Rn+1 = 5Rn + Rn-1, by setting R₁ = 7, R₂ = 21, and for n > 2. 7,21, 112, 518, 3017,... ged(Rn+1, Rn) = 7. (ii) Define an analogous sequence Sn with gcd(Sn+1, Sn) = 23 and justify your answer.
Transcribed Image Text:and the sum for the set of all multiples of k. 1. The Fibonacci numbers are defined by F₁ = 1, F2 = 1, F3 = F₁+F2 = 2, and, in general, for n ≥ 3, Fn = Fn-1 + Fn-2. Thus, the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, .. What is the sum F₁+F3+ + F2n+1? 2. Show that kZ = Prove your answers by induction on n. = { kn\n €Z} (i) 2 Fn Hint: For example in case (ii), write Fn Tn's are related. Here, for k EZ, we will use the notation F₂+F4 + + F2n? 3|n, (ii) 3 F 4|n, (iii) 4Fn 6 | n. So the sequence looks like Prove that - 3. Suppose that a and b are nonzero integers and let = lcm(a, b) be their least common multiple. Show that 3qn+rn with 0≤rn <3 and consider how the 4. Suppose that a₁,..., am are integers, not all 0, and let d = gcd(a₁,...,am). Show that dZ = a₁Z+ a2₂Z + ...+amZ = {a₁₁ + +amum | uj € Z, 1 ≤j≤m}. azn bz=lZ. 5. (i) Define a sequence of numbers Rn, n = 1,2,... Rn+1 = 5Rn + Rn-1, by setting R₁ = 7, R₂ = 21, and for n > 2. 7,21, 112, 518, 3017,... ged(Rn+1, Rn) = 7. (ii) Define an analogous sequence Sn with gcd(Sn+1, Sn) = 23 and justify your answer.
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