The statement "if a = m * n, then the reduced non-negative np factorization is unique" is true.
The following statement is true: if a = m * n, then the reduced non-negative np factorization is unique.
Proof:
Suppose that a = m * n = p * q, where m, n, p, and q are all positive integers. Then, we can write:
m * n = p * q
Dividing both sides by mn, we get:
1 = p / m * q / n
This implies that p / m and q / n are both rational numbers.
Since m and n are coprime, p and q must also be coprime.
Since p / m and q / n are both rational numbers, we can write them as p / m = a / b and q / n = c / d, where a, b, c, and d are all positive integers. Then, we can write:
p = am and q = cn
Substituting these values into the equation a = m * n, we get:
am * cn = m * n
Canceling common factors, we get:
ac = 1
This implies that a and c must be equal to 1 or -1. Since a and c are both positive integers, they must be equal to 1.
Therefore, p = m and q = n, and the reduced non-negative np factorization of a is unique.
Example: Consider the integer a = 12. The prime factorization of 12 is 2^2 * 3. The reduced non-negative np factorization of 12 is (2, 2) * (3, 1).
Now, suppose that a = 12 = 4 * 3. The prime factorization of 4 * 3 is also 2^2 * 3.
The reduced non-negative np factorization of 4 * 3 is also (2, 2) * (3, 1).Therefore, the reduced non-negative np factorization of 12 is unique.