Calculate LCM: Finding the Least Common Multiple of 10 and 15

Q: Why isn't the LCM of 10 and 15 just 10 × 15 = 150?

While $ 10 \times 15 = 150 $ is a common multiple, it's not the least one! Since 10 and 15 share common factors, their LCM is smaller than their product.

Q: How do I know when to stop listing multiples?

Stop as soon as you find the first number that appears in both lists! In this case, 30 appears in both the multiples of 10 and 15, so that's your answer.

Q: Is there a faster way than listing all the multiples?

Yes! You can use the formula: $ \text{LCM} = \frac{a \times b}{\text{GCF}(a,b)} $. For 10 and 15: $ \frac{10 \times 15}{5} = 30 $

Q: Can the LCM ever be smaller than both numbers?

Never! The LCM must be divisible by both original numbers, so it's always greater than or equal to the larger of the two numbers.

LCM Calculation with Multiple Listing

What is the least common multiple of the numbers 10 and 15?

$\boxed {10}~~~\boxed {15 }$

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Step-by-step written solution

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Understand the problem

What is the least common multiple of the numbers 10 and 15?

$\boxed {10}~~~\boxed {15 }$

Step-by-step solution

To find the least common multiple (LCM) of 10 and 15, list the multiples of each number and find the smallest multiple they have in common.

Multiples of 10: $10, 20, 30, 40, \ldots$

Multiples of 15: $15, 30, 45, 60, \ldots$

The smallest common multiple is $30$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: LCM is the smallest positive number divisible by both given numbers
Method: List multiples until you find the first common one: 10, 20, 30...
Verification: Check that 30 ÷ 10 = 3 and 30 ÷ 15 = 2 with no remainders ✓

Common Mistakes

Avoid these frequent errors

Confusing LCM with GCF
Don't find the largest number that divides both = that's GCF, not LCM! LCM is always larger than or equal to both original numbers. Always find the smallest number that both numbers divide into evenly.

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 isn't the LCM of 10 and 15 just 10 × 15 = 150?

While $10 \times 15 = 150$ is a common multiple, it's not the least one! Since 10 and 15 share common factors, their LCM is smaller than their product.

How do I know when to stop listing multiples?

Stop as soon as you find the first number that appears in both lists! In this case, 30 appears in both the multiples of 10 and 15, so that's your answer.

Is there a faster way than listing all the multiples?

Yes! You can use the formula: $\text{LCM} = \frac{a \times b}{\text{GCF}(a,b)}$ . For 10 and 15: $\frac{10 \times 15}{5} = 30$

What if one number is a multiple of the other?

Great question! If one number divides evenly into the other, then the larger number is the LCM. For example, LCM(4, 12) = 12.

Can the LCM ever be smaller than both numbers?

Never! The LCM must be divisible by both original numbers, so it's always greater than or equal to the larger of the two numbers.

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