Given three consecutive positive integers where n is the smallest and is odd, we need to determine the least common multiple (LCM) of these three integers in terms of n.
Let the three consecutive integers be n, n+1, and n+2.
First, let's consider the prime factorizations of these three integers:
- n: Since n is odd, it does not include the prime factor 2.
- n+1: Since n is odd, n+1 is even. Therefore, n+1 includes the prime factor 2.
- n+2: Since n is odd, n+2 is odd and does not include the prime factor 2.
For the least common multiple, we need the highest power of all prime factors that appear in the factorizations of n, n+1, and n+2
LCM:- The prime factor 2 will appear because n+1 is even.
- Any odd prime factors will come from n, n+1, and n+2.
Let's look at the product of n, n+1, and n+2:
n(n+1)(n+2)Since n, n+1, and n+2 are three consecutive numbers, their product includes all necessary factors, but we are interested in the LCM, not just the product.
Simplification:We should check if there is any redundancy in the factors:
- If n or n+2 is a multiple of 3, the highest power of 3 will be included.
- If n is odd, then one of n or n+2 might have additional factors which could reduce to lower powers.
- The number n+1 will include the factor 2 and other odd primes.
Because n is odd and n is not a multiple of 2, n+1 is the only even number contributing the highest power of 2.
Conclusion:The LCM of n, n+1, and n+2 is the product of the highest powers of all prime factors in these three numbers. Given the properties of consecutive numbers, there are no common factors beyond those explicitly present in each number.
Thus, the LCM is:
n(n+1)(n+2)This ensures we include all unique prime factors at their highest powers.