Exercise 2.9. Explain why the division theorem for a = 3 can be restated as follows: For a positive integer b, there exists a nonnegative integer q such that b equals either 3q, 3q + 1, or 3q + 2. Prove this result by induction, following the outline below: 1. First show that 1 can be written in the desired form. 2. Now suppose an integer b that is greater than or equal to 1 can be written in the desired form. Show that b+1 can also be so written. (Hint: There would appear to be three cases here, depending on whether b has the form 3q, 3q+1, or 3q+2, but really the first two can be combined into a single case. First assume thatb has the form 3q or 3g +1 for some nonnegative integer q and show that b+1 can be written in one of the three forms. Then assume that b has the form 3q + 2 for some nonnegative integer q and show that b+1 can be written in one of the three forms.) 3. Conclude by the principle of induction that every positive integer b can be written as 3q, 3q+ 1, or 3q + 2 for some nonnegative integer q. These two examples serve as models for a proof of the division theorem in general.

Exercise 2.9. Explain why the division theorem for a = 3 can be restated as follows: For a positive integer b, there exists a nonnegative integer q such that b equals either 3q, 3q + 1, or 3q + 2. Prove this result by induction, following the outline below: 1. First show that 1 can be written in the desired form. 2. Now suppose an integer b that is greater than or equal to 1 can be written in the desired form. Show that b+1 can also be so written. (Hint: There would appear to be three cases here, depending on whether b has the form 3q, 3q+1, or 3q+2, but really the first two can be combined into a single case. First assume thatb has the form 3q or 3g +1 for some nonnegative integer q and show that b+1 can be written in one of the three forms. Then assume that b has the form 3q + 2 for some nonnegative integer q and show that b+1 can be written in one of the three forms.) 3. Conclude by the principle of induction that every positive integer b can be written as 3q, 3q+ 1, or 3q + 2 for some nonnegative integer q. These two examples serve as models for a proof of the division theorem in general.

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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