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3. Are the following statements true or false? Justify your conclusions. (a) For each integer a, if 3 does not divide a, then 3 divides 2a2 + 1.

3. Are the following statements true or false? Justify your conclusions. (a) For each integer a, if 3 does not divide a, then 3 divides 2a2 + 1.

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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point (a, l (c) Prove that the radius of the circle to the point (a, b) is perpendicular to the line tangent to the circle at the point (a, b). Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to -1. 3. Are the following statements true or false? Justify your conclusions. (a) For each integer a, if 3 does not divide a, then 3 divides 2a2 + 1. (b) For each integer a, if 3 divides 2a2 + 1, then 3 does not divide a. (c) For each integer a, 3 does not divide a if and only if 3 divides 2a2 + 1. 4. Prove that for each real number x and each irrational number q, (x +q) is irrational or (x – q) is irrational. 5. Prove that there exist irrational numbers u and v such that u" is a rational number. Hint: We have proved that v2 is irrational. For the real number q = either q is rational or q is irrational. Use this disjunction to set up two cases.
Transcribed Image Text:point (a, l (c) Prove that the radius of the circle to the point (a, b) is perpendicular to the line tangent to the circle at the point (a, b). Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to -1. 3. Are the following statements true or false? Justify your conclusions. (a) For each integer a, if 3 does not divide a, then 3 divides 2a2 + 1. (b) For each integer a, if 3 divides 2a2 + 1, then 3 does not divide a. (c) For each integer a, 3 does not divide a if and only if 3 divides 2a2 + 1. 4. Prove that for each real number x and each irrational number q, (x +q) is irrational or (x – q) is irrational. 5. Prove that there exist irrational numbers u and v such that u" is a rational number. Hint: We have proved that v2 is irrational. For the real number q = either q is rational or q is irrational. Use this disjunction to set up two cases.
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