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.