LCM of 15 and 21

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 15 and 21. To find the LCM, we need to determine the smallest number that is a multiple of both given numbers.

Answer

$105$

Steps to Solve

Find the prime factorization of each number

Start by determining the prime factors of the numbers 15 and 21.

The prime factorization of 15 is: $$ 15 = 3 \times 5 $$

The prime factorization of 21 is: $$ 21 = 3 \times 7 $$
Identify the highest powers of each prime factor

Identify the unique prime factors from both factorizations and their highest powers.

The prime factors and their highest powers are:
- For 3: The highest power is $3^1$
- For 5: The highest power is $5^1$
- For 7: The highest power is $7^1$
Multiply the highest prime factors together

Now, multiply these highest powers of the prime factors to find the LCM: $$ LCM = 3^1 \times 5^1 \times 7^1 $$

Performing the multiplication: $$ LCM = 3 \times 5 \times 7 $$
Calculate the final result

Carry out the multiplication step-by-step:
- First, calculate $3 \times 5 = 15$
- Then, multiply that result by 7: $15 \times 7 = 105$
So, the least common multiple is: $$ LCM = 105 $$

The least common multiple of 15 and 21 is $105$.

More Information

The least common multiple is useful for finding common denominators in fractions, solving problems involving timed events, and many other applications in math. The LCM of two numbers can often be found using multiple methods, such as listing multiples, prime factorization, or using the relationship between LCM and GCD (greatest common divisor).

Tips

Not finding prime factors carefully: Make sure to get all prime factors correct. Double-check the factorizations.
Forgetting to take the highest power of each prime: It's important to consider the highest power for the LCM, not just the prime factors.
Confusing LCM with GCD: LCM is the least multiple, while GCD is the greatest divisor. Remember they are distinct concepts.

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