Final answer:
You know you have found all the factors of a number when you have tested all integers up to the square root of the number, confirming that they divide the number with no remainder. Factors larger than the square root would have been identified already as quotients with smaller divisors.
Step-by-step explanation:
To know when you have found all of the factors of a number, you must consider each potential factor by dividing the number by integers starting from 1 up to the square root of the number. If the division results in a whole number, then both the divisor and the quotient are factors of the number. Once you have tested all numbers up to and including the square root, you have found all the possible factors. This is because factors of a number larger than its square root would have already appeared as divisors paired with smaller factors. To ensure accuracy, you can check the answer by multiplying the factors in pairs to see if they result in the original number.
For example, to find the factors of 24, we would only test divisors 1 through 4 (the square root of 24 is approximately 4.9). We find that 1, 2, 3, 4, 6, 8, 12, and 24 are the factors because each of these numbers divides 24 without leaving a remainder. Dividing by numbers greater than 4 would be unnecessary as they would not reveal any new factors.