Final answer:
To prove by contradiction that a, b divide n, assume the opposite and show a contradiction. The division theorem states that a = b * q + r, where r is the remainder. Common multiples are integers divisible by both a and b. The proof has implications for the relationship between common multiples and divisors.
Step-by-step explanation:
In order to prove by contradiction that if a positive integer n is a common multiple of a and b, then a, b divide n, we assume the opposite. We assume that a, and b do not divide n. This means that there exists a positive integer k such that n = a * k or n = b * k. However, since a, b is the least common multiple of a and b, it must be a multiple of both a and b. Therefore, this assumption leads to a contradiction, proving that a, and b divide n.
The division theorem states that given positive integers a and b, there exist unique integers q and r such that a = b * q + r, where 0 ≤ r < b. In other words, when we divide a by b, the quotient q is the number of times b can be subtracted from a without going below 0, and the remainder r is the value left after subtracting b * q from a. This theorem is fundamental in understanding division and the properties of multiples.
Common multiples are integers that are divisible by both a and b. These multiples can be found by multiplying a and b by the same positive integer. For example, if a = 3 and b = 4, the common multiples are 12, 24, 36, and so on. Common multiples have several properties: they are always positive, they are multiples of the least common multiple of a and b, and they are infinite, as they can continue indefinitely by multiplying the least common multiple by any positive integer.
The proof by contradiction in this case has important implications. It shows that if a positive integer n is a common multiple of a and b, then a, b divide n. This means that both a and b are factors of n, and any other factors of n must be multiples of a and b as well. Therefore, this proof helps us understand the relationship between common multiples and divisors, providing a deeper understanding of the properties of numbers.