The least common multiple (LCM) of 10 and 12 is the smallest number that is a multiple of both 10 and 12.
Understanding LCM
Before diving into the calculation, let’s understand what LCM represents. Imagine you have two groups of items: one with 10 items and another with 12 items. You want to arrange these items into equal-sized groups without any leftovers. The LCM will give you the size of the largest group possible.
Calculating LCM using Prime Factorization
A common method to find the LCM is through prime factorization.
Factorize each number:
- 10: 2 × 5
- 12: 2^2 × 3
Identify all unique prime factors: 2, 3, 5
For each prime factor, take the highest power it appears in either factorization:
- 2 appears as 2^2 in 12
- 3 appears as 3^1 in 12
- 5 appears as 5^1 in 10
Multiply these highest powers together: LCM = 2^2 × 3 × 5 = 4 × 3 × 5 = 60
Therefore, the LCM of 10 and 12 is 60.
Alternative Method: Listing Multiples
Another way to find the LCM is by listing multiples of each number until you find the smallest common multiple:
- Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, …
- Multiples of 12: 12, 24, 36, 48, 60, 72, 84, …
As you can see, 60 is the first number that appears in both lists.
Real-World Application
Understanding LCM is useful in various real-world scenarios:
Scheduling: If one event occurs every 10 days and another every 12 days, the LCM (60 days) tells you when they will happen on the same day again.
Recipe Scaling: If a recipe serves 10 people and you need to make it for 12, knowing the LCM helps you adjust ingredient quantities proportionally.
Construction: When laying tiles or bricks, knowing the LCM of their dimensions ensures patterns align correctly.
Key Takeaway
The LCM of 10 and 12 is 60. It represents the smallest number that is a multiple of both, and it has practical applications in scheduling, scaling, and pattern alignment.
What’s the difference between LCM and GCF (Greatest Common Factor)?
+The GCF is the largest number that divides both numbers evenly, while the LCM is the smallest number that both numbers divide into evenly. For 10 and 12, the GCF is 2.
Can the LCM be smaller than both numbers?
+No, the LCM is always greater than or equal to the larger of the two numbers. In this case, the LCM (60) is larger than both 10 and 12.
How do I find the LCM of more than two numbers?
+You can use the same prime factorization method, considering the highest power of each prime factor across all numbers.
