What is the LCM of 16 and 18?

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 16 and 18, which is the smallest number that is a multiple of both 16 and 18. To find the LCM, we can use the prime factorization method or find the multiples of the two numbers and determine the smallest one that they both share.

Answer

$144$

Steps to Solve

Find the Prime Factorization of Each Number

First, we need to break down both numbers into their prime factors.

For 16: $$ 16 = 2^4 $$

For 18: $$ 18 = 2^1 \times 3^2 $$
Identify the Highest Power of Each Prime Factor

Next, we will identify the highest power of each prime factor from the factorizations obtained.
- For the prime factor 2, the highest power is $2^4$ (from 16).
- For the prime factor 3, the highest power is $3^2$ (from 18).
Multiply the Highest Powers Together

Now, we can find the LCM by multiplying these highest powers together:

$$ LCM = 2^4 \times 3^2 $$
Calculate the LCM

Finally, we perform the multiplication:
- Calculate $2^4 = 16$
- Calculate $3^2 = 9$
Now, multiply them: $$ LCM = 16 \times 9 $$

Which gives: $$ LCM = 144 $$

The least common multiple (LCM) of 16 and 18 is $144$.

More Information

The least common multiple (LCM) is useful in various applications, such as finding common denominators in fractions and solving problems that involve repeating events. It’s a foundational concept in mathematics that highlights the relation between numbers.

Tips

Forgetting to include each prime factor in the final LCM calculation can lead to an incorrect answer. Make sure to include all prime factors at their highest powers.
Mixing up multiplication order or calculating powers incorrectly may also lead to mistakes. Be careful with arithmetic operations.

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