Find the LCM: Calculating Least Common Multiple of 10 and 15

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

Yes! Use prime factorization: $ 10 = 2 \times 5 $ and $ 15 = 3 \times 5 $. Take the highest power of each prime: $ 2^1 \times 3^1 \times 5^1 = 30 $.

Q: How do I know when I've found the LCM?

The LCM is the first number that appears in both lists of multiples. Once you find it, you can stop listing more multiples!

Q: Can the LCM ever be smaller than both original 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.

Finding LCM with Prime Factorization Method

What is the least common multiple of these two numbers?

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

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Step-by-step written solution

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Understand the problem

What is the least common multiple of these two numbers?

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

Step-by-step solution

To find the least common multiple (LCM) of $10$ and $15$ , we list the multiples of each number:

Multiples of $10$ are $10, 20, 30, 40, \ldots$
Multiples of $15$ are $15, 30, 45, \ldots$

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 GCD
Don't find the greatest common divisor (5) instead of LCM = wrong answer entirely! GCD finds the largest number that divides both, while LCM finds the smallest number both divide into. Always remember LCM is larger than both original numbers.

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

Is there a faster way than listing all multiples?

Yes! Use prime factorization: $10 = 2 \times 5$ and $15 = 3 \times 5$ . Take the highest power of each prime: $2^1 \times 3^1 \times 5^1 = 30$ .

Why isn't 150 the answer since 10 × 15 = 150?

Multiplying the numbers gives you a common multiple, but not necessarily the least one! The LCM is always less than or equal to the product of the two numbers.

How do I know when I've found the LCM?

The LCM is the first number that appears in both lists of multiples. Once you find it, you can stop listing more multiples!

What if one number is a multiple of the other?

Great question! When one number divides evenly into another, the larger number is automatically the LCM. For example, LCM(4, 8) = 8.

Can the LCM ever be smaller than both original 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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