20. Confirm the following properties of the greatest common divisor:(a) If gcd(a , b) = 1, and gcd(a , c) = 1, then gcd(a , bc) = 1.[Hint: Because 1 = ax + by = au + cv for some x, y, u, v,1 = (ax + by)(au + cv) = a(aux + cvx +byu)+bc(yv).](b) If gcd(a , b) = 1, and c | a, then gcd(b , c) = 1.%3D

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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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(1) Given that A = {x : x is even and less than or equal to 10} andB = {x : x is a multiple 3 and less than or equal to 10} in the universe U={1.,2,3….10}Find A, B, AUB, (AUB)’, A∩B, (A∩B)’, A-B, B-A
Students at certain university have to pass of one these three courses (MAT102, STA105, and PHY110). The pass rates for the courses MAT 102, STA105, and PHY110 are 0.54, 0.79, and 0.86, respectively. Suppose that 30 candidates take the MAT102 course, 25 take the STA105 course and 20 take the PHY110 coruse. A student takes one of these courses and passes. What are the probabilities that she took MAT102, STA105 or PHY110? Hint: Denote by Q the event a student to pass their course. Now denote by R1, R2 and R3 the events to take courses MAT102, STA105, and PHY110. The pass reates then represent the conditional probabilites: P(Q| R1), P(Q| R2) and P(Q| R3). The number of students taking the courses can be used to estimate the probabilities for R1, R2 and R3. For example, since there are total of 30 + 25 + 20 = 75 students that take the courses, for R1 we have: P(R1) = 30/75 = 2/5 What you are asked is: given that a student passes (event Q) estimate the probabilities that she took MAT102…
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20. Confirm the following properties of the greatest common divisor: (a) If gcd(a , b) = 1, and gcd(a , c) = 1, then gcd(a , bc) = 1. [Hint: Because 1 = ax + by = au + cv for some x, y, u, v, 1 = (ax + by)(au + cv) = a(aux + cvx + byu)+bc(yv).] (b) If gcd(a , b) = 1, and c|a, then gcd(b , c) = 1.