Final answer:
The smallest positive common multiple of a and b is the least common multiple (LCM). This can be proven using the results on subgroups of Z and the properties of LCM. The LCM is obtained by taking the highest power of each prime factor that appears in either the prime factorization of a or the prime factorization of b.
Step-by-step explanation:
To prove that the smallest positive common multiple of a and b is the least common multiple, we need to use the results on subgroups of Z. Let's assume that m is the smallest positive common multiple of a and b. This means that m is a multiple of both a and b. We can use the properties of subgroups of Z to show that m is also the least common multiple (LCM) of a and b.
To prove this, we need to show that any common multiple of a and b is greater than or equal to m. Suppose there is a common multiple d of a and b such that d < m. Since d is a multiple of a and b, it must also be a multiple of m. But this contradicts the assumption that m is the smallest positive common multiple. Therefore, any common multiple of a and b must be greater than or equal to m, making m the LCM of a and b.
This relationship to prime numbers can be explained by considering the prime factorization of a and b. The LCM of a and b is obtained by taking the highest power of each prime factor that appears in either the prime factorization of a or the prime factorization of b. This ensures that the LCM is divisible by both a and b, and it is the smallest number with this property.
Mathematical induction is not required to prove this statement.