Find the LCM of Denominators: 3, 5, 12, and 15

Prime Factorization with Multiple Numbers

Given four denominators, what is their least common multiple?

$\boxed{3}~~~\boxed{5}~~~\boxed{12}~~~\boxed{15}$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given four denominators, what is their least common multiple?

$\boxed{3}~~~\boxed{5}~~~\boxed{12}~~~\boxed{15}$

Step-by-step solution

To find the least common multiple (LCM) of 3, 5, 12, and 15, we identify their prime factors:

$3 = 3$

$5 = 5$

$12 = 2^2 \times 3$

$15 = 3 \times 5$

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

$2^2 \times 3 \times 5 = 60$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Prime Factorization: Break each number into its prime factors first
Technique: Take highest power of each prime: $2^2 \times 3 \times 5 = 60$
Check: Verify 60 divides by all numbers: 60÷3=20, 60÷5=12, 60÷12=5, 60÷15=4 ✓

Common Mistakes

Avoid these frequent errors

Multiplying all numbers together instead of using prime factorization
Don't just multiply 3×5×12×15 = 2700! This gives a common multiple but not the LEAST common multiple. Always find prime factors first and take the highest power of each prime to get the smallest possible LCM.

Practice Quiz

Test your knowledge with interactive questions

You have a pair of denominators, what is their least common multiple?

$\boxed 2~~~\boxed5$

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, but not the least one! For example, 3×5×12×15 = 2700, but the LCM is only 60. Using prime factorization finds the smallest number that works.

How do I find prime factors of larger numbers like 12?

Start dividing by the smallest primes: 2, 3, 5, 7... For 12: $12 ÷ 2 = 6$ , then $6 ÷ 2 = 3$ , then 3 is prime. So $12 = 2^2 \times 3$ .

What does 'highest power' mean?

Look at each prime across all factorizations. For our example: 2 appears as $2^2$ in 12, 3 appears once in multiple numbers, 5 appears once in 5 and 15. Take the biggest exponent for each.

How can I check if my LCM is correct?

Your LCM should divide evenly into each original number. Test: $60 ÷ 3 = 20$ , $60 ÷ 5 = 12$ , $60 ÷ 12 = 5$ , $60 ÷ 15 = 4$ . All whole numbers means it works!

What if two numbers share the same prime factor?

That's normal! Like 12 and 15 both have factor 3. You only count each prime once in the LCM, using its highest power. This is why the LCM is smaller than just multiplying everything.

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