Final answer:
Gauss' algorithm for integer multiplication has a quadratic time complexity represented as O(n²), meaning its running time grows quadratically with the size of the numbers being multiplied.
Step-by-step explanation:
The question is about the time complexity of Gauss' algorithm for integer multiplication. The running time of an algorithm is often expressed using Big O notation, which provides an upper bound on the time complexity with respect to the input size. The standard algorithm for integer multiplication that we are taught in schools, also known as Gauss' algorithm, has a running time of O(n²). This recurrence indicates the time complexity of the algorithm grows quadratically with the size of the numbers being multiplied.
When multiplying two n-digit numbers using the traditional approach, each digit of the first number is multiplied by every digit of the second number, resulting in n² single-digit multiplication operations. After these multiplications, the results are summed together, which takes additional time, but the dominant factor is still the n² operations. Therefore, the correct answer is 2) O(n²).