Final answer:
The significance of a zero remainder in the Division Algorithm is that the dividend is exactly divisible by the divisor, indicating a factor relationship which is useful in simplifications and fields like cryptography.
Step-by-step explanation:
In the context of the Division Algorithm, when the remainder is zero (i.e., r = 0), it signifies that the dividend is exactly divisible by the divisor. This special case is crucial because it often indicates that the divisor is a factor of the dividend, leading to simplifications in algebra and number theory. For example, in polynomial division, if a polynomial f(x) is exactly divisible by a polynomial g(x), it means that g(x) is a factor of f(x). This can be significant in various mathematical fields, including cryptography, where determining factors is essential.
In the context of the Division Algorithm, when the remainder is zero (r = 0), it signifies that the division is completely divisible without any remainder. This means that the dividend is a multiple of the divisor. For example, if we divide 12 by 3, the remainder is 0, indicating that 12 is divisible by 3. This special case is important because it allows us to determine if one number is a factor of another.