What is the least common multiple (LCM) of 9, 12, and 3?

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 9, 12, and 3. To solve this, we need to find the smallest number that is a multiple of all three given numbers.

Answer

The least common multiple of 9, 12, and 3 is $ 36 $.

Steps to Solve

List the multiples of each number

First, we will find the multiples of each number until we find a common multiple.

Multiples of 9: $$ 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, \ldots $$
Multiples of 12: $$ 12, 24, 36, 48, 60, 72, 84, \ldots $$
Multiples of 3: $$ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, \ldots $$

Identify the common multiples

Next, we look for the smallest number that appears in all three lists of multiples.

The numbers that appear in all lists are: $$36, 72, \ldots$$

The smallest of these is 36.

Determine that 36 is the least common multiple

Since 36 is the smallest number that is a multiple of 9, 12, and 3, we can conclude: $$ \text{LCM}(9, 12, 3) = 36 $$

The least common multiple (LCM) of the numbers 9, 12, and 3 is ( 36 ).

More Information

The least common multiple is useful in various mathematical applications, including adding and subtracting fractions with different denominators. The LCM helps ensure that all fractions are expressed with a common denominator.

Tips

Some common mistakes include:

Not recognizing that all multiples need to be considered, especially if one of the numbers is relatively small.
Confusing the LCM with the greatest common divisor (GCD), which is the largest number that can divide all the given numbers without a remainder.

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