There exist no integers a and b for which 21a+30b = 1. also explain What's a Diophantine equation?

You're correct that there are no integers \( a \) and \( b \) for which \( 21a + 30b = 1 \). This is because the greatest common divisor (GCD) of 21 and 30 is 3, not 1. For any Diophantine equation of the form \( ax + by = d \) to have integer solutions, \( d \) must be divisible by the GCD of \( a \) and \( b \). Since 1 is not divisible by 3, the equation has no integer solutions.

### Diophantine Equation:
A **Diophantine equation** is a type of equation in which only integer solutions are sought. These equations are named after the ancient Greek mathematician Diophantus, who studied them. A linear Diophantine equation has the general form:

\[
ax + by = c
\]

where \( a \), \( b \), and \( c \) are given integers, and \( x \) and \( y \) are the unknowns. The equation has integer solutions if and only if the greatest common divisor (GCD) of \( a \) and \( b \) divides \( c \). In other words, \( \text{GCD}(a, b) \) must be a divisor of \( c \) for integer solutions to exist.

In your example, the GCD of 21 and 30 is 3, and since 1 is not divisible by 3, no integer solution exists for the equation \( 21a + 30b = 1 \).

Would you like more details on how to solve Diophantine equations, or have any other questions?

Here are 5 related questions to think about:
1. How can you find the GCD of two numbers using the Euclidean algorithm?
2. What if the equation was \( 21a + 30b = 3 \)? Would it have solutions?
3. How do you solve a linear Diophantine equation with integer solutions?
4. What are some practical applications of Diophantine equations in number theory?
5. What happens if \( a \) and \( b \) are coprime in a Diophantine equation?

**Tip:** Always check the GCD of the coefficients when determining if a Diophantine equation has integer solutions.

There exist no integers a and b for which 21a + 30b = 1. Also explain what's a Diophantine equation?

Explore the concept of Diophantine equations and learn why there are no integer solutions for the equation 21a + 30b = 1. This problem introduces the GCD theorem and its role in determining the existence of solutions to linear Diophantine equations. Suitable for students in Grades 9-12.

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