Question
Find, with proof, the smallest positive integer m 2 for which both of the following statements are true at the same time: The equation 6x
Find, with proof, the smallest positive integer m 2 for which both of the following statements are true at the same time:
The equation 6x + 11y = m has no non-negative solutions.
For any k > m, the equation 6x + 11y = k has at least one non-negative solution.
The procedure of proof should follow the following format. The answer should like that.
Example
Proposition: For any a,b, if at least one of a,b is not zero, then GCD(a/GCD(a,b), b/GCD(a,b)) = 1
Proof. Let a and b be arbitrary integers.
(1)Since at least one of a,b is non-zero, by definition GCD(a,b) exists and is a positive integer.
(2)By Bzouts Lemma, there exist x and y such that ax + by = GCD(a,b)
(3)By (1), GCD(a,b) is a non-zero integer.
(4)From (2), ax + by = GCD(a,b), and by (3), we can divide both sides by GCD(a,b), so (a/GCD(a,b))x + (b/GCD(a,b))y = 1
(5)By the characterization of GCD, since 1 divides both a/GCD(a,b) and b/GCD(a,b), and from
(4) we know (a/GCD(a,b))x + (b/GCD(a,b))y = 1, it follows that GCD(a/GCD(a,b), b/GCD(a,b)) = 1.
Step by Step Solution
There are 3 Steps involved in it
Step: 1
Get Instant Access to Expert-Tailored Solutions
See step-by-step solutions with expert insights and AI powered tools for academic success
Step: 2
Step: 3
Ace Your Homework with AI
Get the answers you need in no time with our AI-driven, step-by-step assistance
Get Started