Calculate LCM: Finding the Least Common Multiple of 3 and 7

Q: Why is the LCM of 3 and 7 equal to 3 × 7?

Since 3 and 7 are both prime numbers, they share no common factors except 1. When two numbers are relatively prime (GCD = 1), their LCM always equals their product!

Q: Is there a faster way than listing all multiples?

Yes! Use the formula: $ \text{LCM} = \frac{a \times b}{\text{GCD}(a,b)} $. Since GCD(3,7) = 1, we get $ \frac{3 \times 7}{1} = 21 $.

Q: What if I accidentally find a common multiple that's not the least?

That's okay! Any common multiple will be a multiple of the LCM. For example, 42 and 63 are also common multiples of 3 and 7, but 21 is the smallest.

Q: Can the LCM ever be smaller than one of the original numbers?

Never! The LCM must be divisible by each original number, so it's always greater than or equal to the largest number you started with.

Q: How do I check if 21 is really the LCM?

Divide 21 by each original number: $ 21 \div 3 = 7 $ and $ 21 \div 7 = 3 $. Both give whole numbers with no remainders!

LCM with Prime Numbers

What is the least common multiple (LCM) of the numbers 3 and 7?

$\boxed 3~~~\boxed 7$

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Step-by-step written solution

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Understand the problem

What is the least common multiple (LCM) of the numbers 3 and 7?

$\boxed 3~~~\boxed 7$

Step-by-step solution

To find the least common multiple (LCM) of the numbers 3 and 7, we will list the multiples of each number and find the smallest multiple they have in common.

Multiples of 3: $3, 6, 9, 12, 15, 18, 21, \ldots$

Multiples of 7: $7, 14, 21, 28, \ldots$

The smallest common multiple is $21$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: LCM is smallest positive number divisible by all given numbers
Technique: List multiples until finding common one: 3, 6, 9, 12, 15, 18, 21
Check: Verify 21 ÷ 3 = 7 and 21 ÷ 7 = 3 with no remainders ✓

Common Mistakes

Avoid these frequent errors

Confusing LCM with GCD
Don't find the greatest common divisor instead of LCM = getting 1 as the answer! GCD finds the largest factor both numbers share, while LCM finds the smallest multiple both numbers divide into. Always remember LCM is typically larger than both original numbers.

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 is the LCM of 3 and 7 equal to 3 × 7?

Since 3 and 7 are both prime numbers, they share no common factors except 1. When two numbers are relatively prime (GCD = 1), their LCM always equals their product!

Is there a faster way than listing all multiples?

Yes! Use the formula: $\text{LCM} = \frac{a \times b}{\text{GCD}(a,b)}$ . Since GCD(3,7) = 1, we get $\frac{3 \times 7}{1} = 21$ .

What if I accidentally find a common multiple that's not the least?

That's okay! Any common multiple will be a multiple of the LCM. For example, 42 and 63 are also common multiples of 3 and 7, but 21 is the smallest.

Can the LCM ever be smaller than one of the original numbers?

Never! The LCM must be divisible by each original number, so it's always greater than or equal to the largest number you started with.

How do I check if 21 is really the LCM?

Divide 21 by each original number: $21 \div 3 = 7$ and $21 \div 7 = 3$ . Both give whole numbers with no remainders!

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