__Least Common Multiple (LCM or lcm) Notes__

The multiples of a number are what you get when you **multiply it by other numbers** (if you multiply it by 2,3,4,5, etc). Just like the multiplication table.

**Strategy 1**

**List the multiples** of each number (multiply by 2, 3, 4, etc.) then look for the smallest number that appears in each list.

**Example**: Find the lcm for 5, 6, and 15.

- First list the
**multiples**of each number.

Multiples of **5** are 10, 15, 20, 25, **30**, 35, 40,...

Multiples of **6** are 12, 18, 24, **30**, 36, 42, 48,...

Multiples of **15** are **30**, 45, 60, 75, 90,....

- Now, when you look at the list of multiples, you can see that
**30**is the smallest number that appears in each list. So, the least common multiple of 5, 6 and 15 is**30**.

The multiples of 3 are ** 6, 9, 15, ...**,

and the multiples of 5 are

To use this method **factor each of the numbers** into primes. Then for each **different** prime number in all of the factorizations, do the following...

1. **Factor the numbers**

**Count**the number of times each prime number appears in each of the factorizations.- For each prime number, take the largest of these counts.
- Write down that prime number as many times as you counted for it in step 2.

5.The least common multiple is the **product** of all the prime numbers written down.

**Example**: Find the least common multiple of 5, 6 and 15.

**Step 1 - Factor into primes**- Prime factorization of 5 is 5.
- Prime factorization of 6 is
**2 x 3** - Prime factorization of
**15**is**3 x 5** - For each prime number, take the largest of these counts.
- Notice that the
**different primes**are 2, 3 and 5.

**Step 2**-**Count****the number of times each prime number appears in each of the factorizations...**The count of primes in 5 is__one__. The count of primes in 6 is__5____one__and__2____one__. The count of primes in 15 is__3____one__and__3____one____5__**Step 3**-**For each prime number, take the largest of these counts.**The largest count of**2s**is**one.**The largest count of**3s**is**one.**The largest count of**5s**is**one.****Step 4**- Since we now know the count of each prime number, you**write down that prime number as many times as you counted for it in step 3.**

** 2, 3, 5**

**Step 5 - The least common multiple is the****product****of all the prime numbers written down:****2 x 3 x 5 = 30**- The least common multiple of 5, 6 and 15 is
**30**.

After you've calculated a least common multiple, always check to be sure your answer can be divided evenly by both numbers.

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Created by Wanda Collins May 10, 2015 at 1:56pm. Last updated by Wanda Collins May 10, 2015.

Question: Why is this a mathematical limerick?

( (12 + 144 + 20 + 3 Sqrt[4]) / 7 ) + 5*11 = 9^{2} + 0 .

Answer:

A dozen, a gross, and a score,

plus three times the square root of four, divided by seven, plus five times eleven, is nine squared and not a bit more.

---Jon Saxton (math textbook author)

**Presentation Suggestions:**

Challenge students to invent their own math limerick!

**The Math Behind the Fact:**

It is fun to mix mathematics with poetry.

**Resources:**

Su, Francis E., et al. "Math Limerick." Math Fun Facts.

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