Question

True or false (justify your answer):

For all values of A, M, and X_i in the formula
                  X_{i+1} = A ∗ X_i mod M,
      the value of X_{i+1} is always in the range (−M, M)

This is a practice question for a computer science test I have today and I am unsure about the answer? I remember we talked about using this formula in randNumGeneration…? If someone could please help me understand, that would be great. Thanks!

(I think X_{i} is read: X sub i)

Answer 1

X_{i+1} = A ∗ X_i mod M

This means that (i+1)-th value of sequence X is equal to the value of A multiplied by the i-th (previous) value of the X sequence and then fed to modulus operator, which places the result in the [0, M) interval.

So the answer is true.

Example:

let A = 3, M = 5 and X0 = 4.
X1 = A * X0 mod 5 = 3 * 4 mod 5 = 2 mod 5
X2 = A * X1 mod 5 = 3 * 2 mod 5 = 1 mod 5
X3 = A * X2 mod 5 = 3 * 1 mod 5 = 3 mod 5
X4 = A * X3 mod 5 = 3 * 3 mod 5 = 4 mod 5

let A = -3, M = 5 and X0 = 1.
X1 = A * X0 mod 5 = -3 * 1 mod 5 = 2 mod 5
X2 = A * X1 mod 5 = -3 * 2 mod 5 = 4 mod 5
X3 = A * X2 mod 5 = -3 * 4 mod 5 = 3 mod 5
X4 = A * X3 mod 5 = -3 * 3 mod 5 = 1 mod 5