Find the LCM: Calculating Least Common Multiple of 7, 11, and 13

Q: Why can I just multiply prime numbers together?

Since prime numbers have no common factors except 1, the only way to create a number divisible by all of them is to multiply them together. There's no smaller number that works!

Q: What if the numbers weren't all prime?

For non-prime numbers, you'd need to find the prime factorization of each number first, then use the highest power of each prime factor. But with all primes, multiplication is the shortcut!

Q: How do I know 7, 11, and 13 are prime?

Prime numbers are only divisible by 1 and themselves. Check: 7 ÷ 2, 7 ÷ 3, 7 ÷ 5 don't give whole numbers. Same pattern works for 11 and 13!

Q: Could the LCM be smaller than 1001?

No! The LCM must be divisible by all three numbers. Since 7, 11, and 13 are prime, any common multiple must contain all three as factors, making 1001 the smallest possibility.

Q: Is there a pattern for three prime numbers?

Yes! For any set of distinct prime numbers, their LCM is always their product. This makes prime number LCM problems much easier than composite number problems.

Prime Number LCM with Direct Multiplication

What is the least common multiple of the following numbers?

$\boxed{7} ~~~ \boxed{11} ~~~ \boxed{13}$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

What is the least common multiple of the following numbers?

$\boxed{7} ~~~ \boxed{11} ~~~ \boxed{13}$

Step-by-step solution

To find the least common multiple (LCM) of the numbers 7, 11, and 13, we first recognize that all these numbers are prime. The LCM is given by multiplying these numbers together.

The formula for the LCM of three numbers $a, b, c$ is:

$\text{LCM}(a, b, c) = a \times b \times c$

Substituting into the formula gives:

$\text{LCM}(7, 11, 13) = 7 \times 11 \times 13$

Calculating the product:

$7 \times 11 = 77$

$77 \times 13 = 1001$

So, the least common multiple of 7, 11, and 13 is 1001.

Final Answer

1001

Key Points to Remember

Essential concepts to master this topic

Prime Rule: When all numbers are prime, LCM equals their product
Technique: Calculate step by step: 7 × 11 = 77, then 77 × 13 = 1001
Check: Verify 1001 ÷ 7 = 143, 1001 ÷ 11 = 91, 1001 ÷ 13 = 77 ✓

Common Mistakes

Avoid these frequent errors

Using addition instead of multiplication for prime numbers
Don't add prime numbers like 7 + 11 + 13 = 31 to find LCM! This gives a number that's not divisible by all original numbers. Always multiply all prime numbers together since they share no common factors.

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 I just multiply prime numbers together?

Since prime numbers have no common factors except 1, the only way to create a number divisible by all of them is to multiply them together. There's no smaller number that works!

What if the numbers weren't all prime?

For non-prime numbers, you'd need to find the prime factorization of each number first, then use the highest power of each prime factor. But with all primes, multiplication is the shortcut!

How do I know 7, 11, and 13 are prime?

Prime numbers are only divisible by 1 and themselves. Check: 7 ÷ 2, 7 ÷ 3, 7 ÷ 5 don't give whole numbers. Same pattern works for 11 and 13!

Could the LCM be smaller than 1001?

No! The LCM must be divisible by all three numbers. Since 7, 11, and 13 are prime, any common multiple must contain all three as factors, making 1001 the smallest possibility.

Is there a pattern for three prime numbers?

Yes! For any set of distinct prime numbers, their LCM is always their product. This makes prime number LCM problems much easier than composite number problems.

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