Find the LCM of Denominators: 9, 11, and 13

Q: How do I know if I should multiply the numbers or use a different method?

First check if the numbers share any common factors. Since 9, 11, and 13 are coprime (share no factors except 1), you simply multiply them together!

Q: What if the numbers weren't coprime?

Use prime factorization! Write each number as a product of primes, then take the highest power of each prime that appears. For example: LCM(12,18) = LCM(2²×3, 2×3²) = 2²×3² = 36.

Q: Why is 1287 the smallest number divisible by 9, 11, and 13?

Since these numbers have no common factors, any common multiple must include all their prime factors. The smallest such number is their product: $ 9 \times 11 \times 13 = 1287 $.

Q: How can I check my LCM is correct?

Divide your LCM by each original number. If you get whole numbers every time, your LCM is correct! Try: 1287÷9=143, 1287÷11=117, 1287÷13=99.

LCM with Prime Number Factors

Given several denominators, what is their least common multiple?

$\boxed{9} \boxed{11} \boxed{13}$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given several denominators, what is their least common multiple?

$\boxed{9} \boxed{11} \boxed{13}$

Step-by-step solution

The least common multiple (LCM) of $9, 11, \text{ and } 13$ is the smallest positive integer that is divisible by each of these numbers.

Since there are no common factors other than 1, the LCM is simply the product of these numbers:

$9 \times 11 \times 13$ equals 1287.

The LCM is $1287$ .

Final Answer

1287

Key Points to Remember

Essential concepts to master this topic

Rule: When numbers share no common factors, LCM equals their product
Technique: Check if numbers are coprime: $9 \times 11 \times 13 = 1287$
Check: Verify 1287 divides by each: 1287÷9=143, 1287÷11=117, 1287÷13=99 ✓

Common Mistakes

Avoid these frequent errors

Adding denominators instead of finding LCM
Don't add 9+11+13=33 as the answer! This gives a number too small to be divisible by all three denominators. Always multiply coprime numbers or use prime factorization to find the true LCM.

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

How do I know if I should multiply the numbers or use a different method?

First check if the numbers share any common factors. Since 9, 11, and 13 are coprime (share no factors except 1), you simply multiply them together!

What if the numbers weren't coprime?

Use prime factorization! Write each number as a product of primes, then take the highest power of each prime that appears. For example: LCM(12,18) = LCM(2²×3, 2×3²) = 2²×3² = 36.

Why is 1287 the smallest number divisible by 9, 11, and 13?

Since these numbers have no common factors, any common multiple must include all their prime factors. The smallest such number is their product: $9 \times 11 \times 13 = 1287$ .

How can I check my LCM is correct?

Divide your LCM by each original number. If you get whole numbers every time, your LCM is correct! Try: 1287÷9=143, 1287÷11=117, 1287÷13=99.

Is there a faster way to solve this?

For small coprime numbers like these, multiplication is fastest! But for larger numbers, try listing multiples or using the formula: LCM(a,b) = (a×b)÷GCD(a,b).

What does this have to do with adding fractions?

When adding fractions with denominators 9, 11, and 13, you need a common denominator. The LCM (1287) is the smallest denominator that works for all three fractions!

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