Given equation: 121x - 1331y = 100

To find the number of solutions for this equation, we need to analyze the coefficients of x and y and see if there is a common factor between them.

Greatest Common Divisor (GCD):
The GCD of 121 and 1331 is 11. Since 11 is a common factor between the coefficients, we can rewrite the equation as:

11(11x - 121y) = 100

Divisibility Test:
For an equation of the form ax + by = c to have integer solutions, the right-hand side (c) must be divisible by the GCD of a and b. In this case, the right-hand side is 100, which is not divisible by 11.

Conclusion:
Since 100 is not divisible by 11, the equation 121x - 1331y = 100 does not have any integer solutions.

Alternate Explanation:
Another way to see why the equation has no integer solutions is by considering the coefficients of x and y.

121x - 1331y = 100

The coefficient of x is 121, which is not divisible by 11. Therefore, for any integer value of x, the left-hand side of the equation will not be divisible by 11.

Similarly, the coefficient of y is 1331, which is also not divisible by 11. Therefore, for any integer value of y, the left-hand side of the equation will not be divisible by 11.

Since neither the coefficient of x nor the coefficient of y is divisible by 11, it is impossible for the left-hand side of the equation to be equal to 100, which means there are no integer solutions for this equation.

Final Answer:
The number of solutions of the equation 121x - 1331y = 100, where both x and y are integers, is 0.