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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.

### Example: Find the least common multiple for 3 and 5:

The multiples of 3 are 6, 9, 15, ...,
and the multiples of 5 are 10, 15, 20, ..., like this:

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

1. Count the number of times each prime number appears in each of the factorizations.
2. For each prime number, take the largest of these counts.
3. 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 5.  The count of primes in 6 is one 2 and one 3.  The count of primes in 15 is one 3 and 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.

Least Common Multiple Activity

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## Notes

### Figure This Challenge #56

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

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# Math Limerick

Question: Why is this a mathematical limerick?

( (12 + 144 + 20 + 3 Sqrt) / 7 ) + 5*11 = 92 + 0 .

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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