Find the LCM of Denominators: 5, 6, and 15 - Step-by-Step Solution

Q: Why can't I just multiply 5 × 6 × 15 to get the answer?

While 450 is a common multiple, it's not the least common multiple! Numbers often share factors - like how 6 and 15 both contain the factor 3. The LCM uses each prime factor only once at its highest power.

Q: What if one number is a factor of another?

Great observation! Since 5 divides into 15, the LCM will be at least 15. You only need the highest power of each prime, so 5 is already 'covered' by 15 in this problem.

Q: How do I know when I have the right prime factorization?

Keep dividing by the smallest prime possible until you can't divide anymore. For 6: 6 ÷ 2 = 3, then 3 ÷ 3 = 1. So $ 6 = 2 \times 3 $.

Q: What if the numbers have repeated prime factors?

Take the highest power of each prime! For example, if you had 12 = $ 2^2 \times 3 $ and 8 = $ 2^3 $, you'd use $ 2^3 $ (the higher power) in your LCM.

Q: Can I check my LCM answer?

Always check! Your LCM should divide evenly by each original number. For 30: 30 ÷ 5 = 6 ✓, 30 ÷ 6 = 5 ✓, and 30 ÷ 15 = 2 ✓. All give whole numbers!

LCM with Prime Factorization Method

Given several denominators, what is their least common multiple?

$\boxed5~~~\boxed6 ~~~\boxed{15}$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given several denominators, what is their least common multiple?

$\boxed5~~~\boxed6 ~~~\boxed{15}$

Step-by-step solution

To find the least common multiple (LCM) of the denominators $5, 6, 15$ , we need to consider each prime factor of these numbers at their highest power:

$5$ : prime itself
$6 = 2 \times 3$
$15 = 3 \times 5$

Therefore, the LCM is:

$2 \times 3 \times 5 = 30$

So, the least common multiple of $5, 6, 15$ is $30$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break each number into prime factors first
Technique: Use highest power of each prime: $2^1 \times 3^1 \times 5^1 = 30$
Check: Verify 30 divides evenly by 5, 6, and 15 ✓

Common Mistakes

Avoid these frequent errors

Multiplying all numbers together instead of finding LCM
Don't just multiply 5 × 6 × 15 = 450! This gives a common multiple, but not the least one. The LCM is much smaller because numbers share common factors. Always use prime factorization to find the smallest common multiple.

Practice Quiz

Test your knowledge with interactive questions

Without calculating, determine whether the quotient in the division exercise is less than 1 or not:

$$ 5:6= $$

FAQ

Everything you need to know about this question

Why can't I just multiply 5 × 6 × 15 to get the answer?

While 450 is a common multiple, it's not the least common multiple! Numbers often share factors - like how 6 and 15 both contain the factor 3. The LCM uses each prime factor only once at its highest power.

What if one number is a factor of another?

Great observation! Since 5 divides into 15, the LCM will be at least 15. You only need the highest power of each prime, so 5 is already 'covered' by 15 in this problem.

How do I know when I have the right prime factorization?

Keep dividing by the smallest prime possible until you can't divide anymore. For 6: 6 ÷ 2 = 3, then 3 ÷ 3 = 1. So $6 = 2 \times 3$ .

What if the numbers have repeated prime factors?

Take the highest power of each prime! For example, if you had 12 = $2^2 \times 3$ and 8 = $2^3$ , you'd use $2^3$ (the higher power) in your LCM.

Can I check my LCM answer?

Always check! Your LCM should divide evenly by each original number. For 30: 30 ÷ 5 = 6 ✓, 30 ÷ 6 = 5 ✓, and 30 ÷ 15 = 2 ✓. All give whole numbers!

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