For Theorem 2.8 and it corollary give a simplified and short example in order to explain it very well as a presentation.Ps(short and well structured) As an example, the positive common multiples of the integers -12 and30, 120, 180, ... ; hence, lcm(-12, 30) = 60.The following remark is clear from our discussion: given nonzero integers%3Dand b, lcm(a, b) always exists and Icm(a, b) < |ab|.We lack a relationship between the ideas of greatest common divisor andcommon multiple. This gap is filled by Theorem 2.8.Theorem 2.8. For positive integers a and bgcd(a, b) lcm(a, b) = abProof. To begin, put d = gcd(a, b) and write a = dr,b= ds for integers r and s.m = ab/d, then m = as = rb, the effect of which is to make m a (positive) commonmultiple of a and b.Now let c be any positive integer that is a common multiple of a and b; say,for definiteness, c = au = bv. As we know, there exist integers x and y ratisfyingd = ax + by. In consequence,%3Dcdc(ax + by);)* * (E):y = vx +uymababThis equation states that m | c, allowing us to conclude that m < c. Thus, in accordancewith Definition 2.4, m = lcm(a, b); that is,ablcm(a, b) =dabgcd(a, b)which is what we started out to prove.Theorem 2.8 has a corollary that is worth a separate statement.Corollary. For any choice of positive integers a and b, Icm(a, b) = ab if and only ifgcd(a, b) = 1.Perhaps the chief virtue of Theorem 2.8 is that it makes the calculation of the=ast common multiple of two integers dependent on the value of their greatestommon divisor-which, in turn, can be calculated from the Euclidean Algorithmhen considering the positive integers 3054 and 12378, for instance, we found theted(3054, 12378) = 6; whence,3054 12378lcm(3054, 12378) == 63004026.Before moving on to other matters, let us observe that the notion ofommon divisor can he extendod t

Answered: Image /qna-images/answer/68591835-efde-46f3-b380-115f6261d0f9.jpg

Answered: For Theorem 2.8 and it corollary give a…

For Theorem 2.8 and it corollary give a simplified and short example in order to explain it very well as a presentation. Ps(short and well structured)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

For DAY 1 Evaluate each relation for (a) x = y² +3 (b) y = x² + x 3 (c) y = x ²³² + x 17. x-axis, y-axis, orign symmetry 1-17 are practice for Midterm DAY 1
I wanted to know why this is false? I thought if you plug in any positive or negative # in x you get a # above -1 and less than infinity?
The similarity between an ordinal level and an interval level is that.
Not sure how to continue the problem. The red color is my work . Am I going in the right direction?
Part three was incorect. I was told you cannot do the computatiopns without substituting in the T and F Values first. Can you do show me how to do it that way?
QUESTION 2 Jim wants to buy a computer. The total cost is $1,180. If he can save $60 a month, how long will it take for him to save up for the computer? For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). BIUS Paragraph Arial 14px !!
Can you please answer this question and show all of the steps and say the correct letter
Please help me with this, thank-you!
2. A power line runs from A(-4,-5) to (6, 1). The City is running another power line, and it must be the right bisector of AB. Determine the equation for the right bisector of AB. Express the equation is in standard form. Include a sketch and label it. Remember, the sketch is to help you visualize the problem. Do NOT use the diagram to determine important information!! You must do the calculations!!