Find the LCM of Denominators: Calculating LCM(10, 15, 25)

Q: What's the difference between LCM and GCF?

LCM finds the smallest number that all given numbers divide into. GCF finds the largest number that divides into all given numbers. They're opposites!

LCM with Prime Factorization Method

Given several denominators, what is their least common multiple?

$\boxed{10}~~~\boxed{15} ~~~\boxed{25}$

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Step-by-step written solution

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Understand the problem

Given several denominators, what is their least common multiple?

$\boxed{10}~~~\boxed{15} ~~~\boxed{25}$

Step-by-step solution

To find the least common multiple (LCM) of $10$ , $15$ , and $25$ , we first find their prime factorizations:

$10 = 2 \, \times \, 5$

$15 = 3 \, \times \, 5$

$25 = 5^2$

The LCM is found by multiplying the highest powers of every prime number: $2^1$

LCM = $2 \, \times \, 3 \, \times \, 25 = 6 \, \times \, 25 = 150$ .

Final Answer

150

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: $2^1 \times 3^1 \times 5^2 = 150$
Check: Verify 150 ÷ 10 = 15, 150 ÷ 15 = 10, 150 ÷ 25 = 6 (all whole) ✓

Common Mistakes

Avoid these frequent errors

Multiplying all numbers together without considering prime factors
Don't just multiply 10 × 15 × 25 = 3750! This gives you a common multiple but not the LEAST common multiple. Always find prime factorizations and use only the highest power of each prime factor.

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 the numbers together?

Multiplying all numbers gives you a common multiple (3750), but not the smallest one. The LCM method finds the most efficient common multiple by avoiding unnecessary repeated factors.

What if I forget to use the highest power of a prime?

You'll get a number that's too small and won't be divisible by all original numbers. For example, using $5^1$ instead of $5^2$ gives 30, but 30 ÷ 25 isn't a whole number!

How do I find prime factorizations quickly?

Start with the smallest primes (2, 3, 5, 7...) and divide repeatedly:

10 = 2 × 5
15 = 3 × 5
25 = 5 × 5 = $5^2$

What's the difference between LCM and GCF?

LCM finds the smallest number that all given numbers divide into. GCF finds the largest number that divides into all given numbers. They're opposites!

Can I check my LCM answer?

Yes! Divide your LCM by each original number. If all results are whole numbers, your LCM is correct. For 150: 150÷10=15, 150÷15=10, 150÷25=6 ✓

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