Make a general statement about the LCM for the following pairs of numbers. You could consider examples before coming up with these general statements. Look for possible explanations of why they hold. (a) Two multiples of 3 (b) Two consecutive even numbers (c) Two consecutive numbers (d) Two co-prime numbers
LCM of two multiples of 3
LCM is a multiple of 3. If numbers are 3a and 3b, LCM = 3 × LCM(a, b).
LCM of two consecutive even numbers
LCM = product ÷ 2. For 2n and 2(n+1), LCM = 2n(n+1).
LCM of two consecutive numbers
LCM = product of the two numbers.
LCM of two co-prime numbers
LCM = product of the two numbers.
Explanation
(a) Both numbers share a factor of 3, so LCM must be a multiple of 3. (b) HCF of consecutive even numbers is 2, so LCM = product ÷ HCF = product ÷ 2. (c) Consecutive numbers are co-prime (HCF = 1), so LCM = product. (d) By definition, co-prime numbers have HCF = 1, so LCM = product.