Question

Are there \(s, t \in Z\) such that \(24 s+14 t=17\)

Short answer

No, such integers do not exist.

Step-by-step solution

Assess the equation setup

We are given the equation \(24s + 14t = 17\) and need to determine if there exist integers \(s\) and \(t\) such that this equation holds true.

Find the Greatest Common Divisor (GCD)

Calculate the greatest common divisor of the coefficients of \(s\) and \(t\), which are 24 and 14. Using the Euclidean algorithm, we find \(\gcd(24, 14)\): - \(24 \div 14\) has a remainder of 10.- \(14 \div 10\) has a remainder of 4.- \(10 \div 4\) has a remainder of 2.- \(4 \div 2\) has a remainder of 0.Thus, the GCD is 2.

Check for divisibility condition

For integers \(s\) and \(t\) to exist, the GCD of 24 and 14 must divide 17. Since \(2 mid 17\) (2 does not divide 17), there are no integers \(s\) and \(t\) that satisfy this equation.

Key concepts

Greatest Common Divisor (GCD)

The concept of the Greatest Common Divisor, often abbreviated as GCD, is fundamental in number theory. The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. For example, when considering the numbers 24 and 14, their common divisors are 1, 2, and 7, with 2 being the largest. Finding the GCD is crucial when simplifying fractions and solving equations like Diophantine equations.

The GCD helps us understand the relationship between two numbers and whether their ratio can be simplified further. Learning to calculate the GCD allows for simplification of problems involving divisibility and congruence theory, making it easier to solve equations involving integer solutions.

Euclidean algorithm

The Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers. It relies on the principle that the GCD of two numbers also divides their difference. This algorithm iteratively replaces the larger number by its remainder when divided by the smaller number, until one of the numbers is zero. The non-zero number at this point is the GCD.

In the exercise example, we calculated the GCD of 24 and 14 using this algorithm as follows:

• Divide 24 by 14, giving a remainder of 10.
• Next, divide 14 by 10, leaving a remainder of 4.
• Then, divide 10 by 4, with a remainder of 2.
• Finally, 4 divided by 2 leaves no remainder, ending the algorithm with 2 as the GCD.

This systematic approach is not only quick but also reduces the possibility of errors, which can arise when listing divisors manually.

Divisibility condition

The divisibility condition is a key concept when determining if a Diophantine equation has integer solutions. For the equation to have solutions for integers, the constant term on the right must be divisible by the GCD of the coefficients of the integer variables on the left side.

In our problem, the equation is 24s + 14t = 17. After finding that the GCD of 24 and 14 is 2, we assess whether 17 is divisible by 2 to check the possibility of integer solutions. Since 17 divided by 2 does not yield an integer, there are no integers s and t such that 24s+14t equals 17.

This condition serves as a quick check: if the divisibility requirement is not met, there's no need to even attempt to find specific integer solutions, saving time and effort in problem-solving.