Lesson 27: Multiples, Least Common Multiple, Equivalent Division Problems

New Concept

The least common multiple is the smallest number that is a multiple of a group of numbers. The term least common multiple is often abbreviated LCM.

What’s next

Now you have the foundation. Next, you'll see worked examples for finding the LCM by listing multiples and by using prime factorization.

Property

The multiples of a number are produced by multiplying the number by 1, by 2, by 3, by 4, and so on. A number appearing in the lists of two different numbers is a common multiple.

Examples

• The multiples of 5 are: $5, 10, 15, 20, 25, 30, ...$
• The multiples of 8 are: $8, 16, 24, 32, 40, 48, ...$
• The common multiples of 4 and 6 are $12, 24, 36, ...$ because they appear in both lists.

Explanation

Think of multiples as skip-counting! To find the multiples of a number, you just keep adding that number to itself. It is like building a ladder where each rung is one more group of that number. A common multiple is a number that shows up in the lists for two different numbers, like a landmark they both pass.

Property

The least common multiple is the smallest number that is a multiple of two or more numbers. The term least common multiple is often abbreviated LCM.

Examples

• For 6 and 8, check multiples of 8: $8, 16, 24$. Since 24 is divisible by 6, the LCM of 6 and 8 is 24.
• For 3, 4, and 6, check multiples of 6: Is 6 divisible by 4? No. Is 12 divisible by 4 and 3? Yes! The LCM is 12.
• For 8 and 10, check multiples of 10: $10, 20, 30, 40$. Since 40 is divisible by 8, the LCM of 8 and 10 is 40.

Explanation

The LCM is the very first number you hit that is on everyone’s “multiples” list. Instead of listing out endless numbers, just check the multiples of the biggest number in your set. See if the other numbers divide into it evenly. It is the ultimate shortcut to finding the first spot where all their paths cross!

Property

The LCM of a set of numbers is the product of all the prime factors necessary to form any number in the set.

Examples

• For 18 and 24: $18 = 2 \cdot 3 \cdot 3$ and $24 = 2 \cdot 2 \cdot 2 \cdot 3$. The LCM needs three 2s and two 3s, so $\operatorname{LCM} = 2^3 \cdot 3^2 = 72$.
• For 24 and 40: $24 = 2^3 \cdot 3$ and $40 = 2^3 \cdot 5$. The LCM needs three 2s, one 3, and one 5, so $\operatorname{LCM} = 2^3 \cdot 3 \cdot 5 = 120$.
• For 30 and 75: $30 = 2 \cdot 3 \cdot 5$ and $75 = 3 \cdot 5^2$. The LCM needs one 2, one 3, and two 5s, so $\operatorname{LCM} = 2 \cdot 3 \cdot 5^2 = 150$.

Explanation

Break down each number into its prime building blocks, like LEGOs! To find the LCM, you need to gather the maximum number of each prime factor found in any of the original numbers. It is like making sure you have all the necessary pieces to build any of the numbers, then multiplying them together for the smallest super-combo.