Find the LCM: Calculating Least Common Multiple of 2 and 5

Q: Why isn't the LCM of 2 and 5 equal to 2 × 5?

Since 2 and 5 are co-prime (share no common factors except 1), their LCM equals their product: $ 2 \times 5 = 10 $. This only works when numbers have no common factors!

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

LCM (Least Common Multiple) is the smallest number both divide into, while GCD (Greatest Common Divisor) is the largest number that divides both. For 2 and 5: LCM = 10, GCD = 1.

Q: How do I know when to stop listing multiples?

Stop when you find the first number that appears in both lists. For 2 and 5, you see 10 appears in both sequences, so that's your LCM!

Q: Can the LCM be smaller than both numbers?

No! The LCM must be divisible by both numbers, so it's always greater than or equal to the larger number. In this case, LCM(2,5) = 10 > 5.

LCM with Co-prime Numbers

What is the least common multiple of these two numbers?

$\boxed{2}~~~\boxed{5}$

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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 these two numbers?

$\boxed{2}~~~\boxed{5}$

Step-by-step solution

To find the least common multiple (LCM) of $2$ and $5$ , we list the multiples of each number:

Multiples of $2$ are $2, 4, 6, 8, 10, \ldots$
Multiples of $5$ are $5, 10, 15, \ldots$

The smallest common multiple is $10$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: LCM is smallest positive integer divisible by both numbers
Technique: List multiples: 2 → 2,4,6,8,10... and 5 → 5,10,15...
Check: Verify 10 ÷ 2 = 5 and 10 ÷ 5 = 2 ✓

Common Mistakes

Avoid these frequent errors

Adding the two numbers instead of finding common multiples
Don't simply add 2 + 5 = 7! Addition doesn't find shared multiples, so 7 isn't divisible by both numbers. Always list actual multiples of each number and find the smallest one they share.

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 isn't the LCM of 2 and 5 equal to 2 × 5?

Since 2 and 5 are co-prime (share no common factors except 1), their LCM equals their product: $2 \times 5 = 10$ . This only works when numbers have no common factors!

What's the difference between LCM and GCD?

LCM (Least Common Multiple) is the smallest number both divide into, while GCD (Greatest Common Divisor) is the largest number that divides both. For 2 and 5: LCM = 10, GCD = 1.

How do I know when to stop listing multiples?

Stop when you find the first number that appears in both lists. For 2 and 5, you see 10 appears in both sequences, so that's your LCM!

Can the LCM be smaller than both numbers?

No! The LCM must be divisible by both numbers, so it's always greater than or equal to the larger number. In this case, LCM(2,5) = 10 > 5.

What if I need the LCM of more than two numbers?

Use the same method! List multiples of each number and find the smallest one that appears in all lists. You can also find LCM step by step: LCM(a,b,c) = LCM(LCM(a,b),c).

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