Question

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) There exist integers \(a\) and \(b\) for which \(42 a+7 b=1\).

Short answer

The statement is false. There are no integers \(a\) and \(b\) that can satisfy the equation \(42a + 7b = 1\), because the greatest common divisor of 42 and 7 is 7, not 1.

Step-by-step solution

Identifying the GCD

First, the greatest common divisor (GCD) of 42 and 7 should be identified. The GCD of 42 and 7 is 7, because 7 is the largest number that can divide both 42 and 7 without a remainder.

Applying Bezout's identity

Bezout's identity states that for any two integers \(a, b\) not both zero, there exist integers \(x, y\) such that \(ax + by = gcd(a, b)\). Our equation is \(42a + 7b = 1\), but the gcd of 42 and 7 is 7 not 1, so there is no solution in integers for this equation given that Bezout's Identity holds.

Final Statement

Based on Bezout's Identity, which ensures the existence of a linear combination of given numbers that forms their gcd, the statement is false because there are no integers for \(a\) and \(b\) that satisfy the equation \(42a + 7b = 1\) since the gcd of 42 and 7 is 7, not 1.