Find the LCM: Calculating Least Common Multiple of 18, 24, and 30

Q: Why can't I just multiply all three numbers together?

Multiplying 18 × 24 × 30 = 12,960 gives you a common multiple, but not the least common multiple! The LCM method finds the smallest number that works.

Q: How do I know which power of each prime to use?

Look at each prime factor across all numbers and pick the highest power. For example, since 24 has $ 2^3 $, use $ 2^3 $ in your LCM, not $ 2^1 $ from the other numbers.

Q: Can I use a different method besides prime factorization?

Yes! You can use the listing method (write multiples of each number until you find a match) or division method, but prime factorization is usually fastest for larger numbers.

Q: What if two numbers share the same prime factors?

That's common! Just remember to use the highest power of each shared prime. For 18 = $ 2^1 \times 3^2 $ and 24 = $ 2^3 \times 3^1 $, use $ 2^3 $ and $ 3^2 $.

LCM Calculation with Multiple Prime Factors

Among these numbers, what is the least common multiple?

$\boxed{18}~~~\boxed{24} ~~~\boxed{30}$

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Step-by-step written solution

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Understand the problem

Among these numbers, what is the least common multiple?

$\boxed{18}~~~\boxed{24} ~~~\boxed{30}$

Step-by-step solution

To find the least common multiple (LCM) of $18$ , $24$ , and $30$ , first find their prime factorizations:

$18 = 2 \, \times \, 3^2$

$24 = 2^3 \, \times \, 3$

$30 = 2 \, \times \, 3 \, \times \, 5$

The LCM is obtained by using the highest power of each prime:

$2^3$ from 24, $3^2$ from 18, and $5^1$ from 30.

The LCM is $2^3 \, \times \, 3^2 \, \times \, 5 = 8 \, \times \, 9 \, \times \, 5 = 360$ .

Final Answer

360

Key Points to Remember

Essential concepts to master this topic

Prime Factorization Rule: Find all prime factors for each number first
Highest Power Method: Use $2^3 \times 3^2 \times 5^1 = 360$ from 18, 24, 30
Verification Check: Confirm 360 ÷ 18 = 20, 360 ÷ 24 = 15, 360 ÷ 30 = 12 ✓

Common Mistakes

Avoid these frequent errors

Adding prime factors instead of using highest powers
Don't add exponents like 2¹ + 2³ = 2⁴ = 16 in your LCM calculation! This gives wrong answers like 720 instead of 360. Always use the highest power of each prime factor that appears in any number.

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 all three numbers together?

Multiplying 18 × 24 × 30 = 12,960 gives you a common multiple, but not the least common multiple! The LCM method finds the smallest number that works.

What if I forget to include a prime factor?

Missing a prime factor means your answer won't be divisible by all the original numbers. Always check that every prime from every number appears in your LCM.

How do I know which power of each prime to use?

Look at each prime factor across all numbers and pick the highest power. For example, since 24 has $2^3$ , use $2^3$ in your LCM, not $2^1$ from the other numbers.

Can I use a different method besides prime factorization?

Yes! You can use the listing method (write multiples of each number until you find a match) or division method, but prime factorization is usually fastest for larger numbers.

What if two numbers share the same prime factors?

That's common! Just remember to use the highest power of each shared prime. For 18 = $2^1 \times 3^2$ and 24 = $2^3 \times 3^1$ , use $2^3$ and $3^2$ .

How can I check if 360 is really the LCM?

Divide 360 by each original number: 360÷18=20, 360÷24=15, 360÷30=12. If all results are whole numbers, then 360 is divisible by all three numbers! ✓

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