Find the LCM of Denominators: Calculating LCM(7,5)

Q: Why is the LCM of 7 and 5 equal to 7 × 5?

Because 7 and 5 are coprime - they share no common factors except 1. When numbers have no common factors, their LCM always equals their product: $ 7 \times 5 = 35 $.

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

LCM finds the smallest number both denominators divide into, while GCD finds the largest number that divides both. For coprime numbers like 7 and 5: LCM = 35, GCD = 1.

Q: Do I always need to list all the multiples?

Not always! For coprime numbers (like 7 and 5), you can simply multiply them. For numbers with common factors, use prime factorization or the listing method.

Q: How do I know when I've found the LCM?

The LCM is the first number that appears in both lists of multiples. In this case, 35 is the first common multiple of both 7 and 5.

Q: Why can't I just use 70 or 105 as the common multiple?

You could use 70 or 105, but we want the least common multiple. Using 35 instead of larger multiples makes fraction calculations much easier!

LCM Calculation with Coprime Numbers

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

$\boxed 7~~~\boxed 5$

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Step-by-step written solution

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Understand the problem

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

$\boxed 7~~~\boxed 5$

Step-by-step solution

To determine the least common multiple (LCM) of $7$ and $5$ , we list the multiples of each number.

Multiples of $7$ : $7, 14, 21, 28, 35, \ldots$

Multiples of $5$ : $5, 10, 15, 20, 25, 30, 35, \ldots$

The smallest common multiple is $35$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

LCM Rule: For coprime numbers, LCM equals their product
Method: List multiples: 7: 7,14,21,28,35... and 5: 5,10,15,20,25,30,35...
Verify: Check that 35 ÷ 7 = 5 and 35 ÷ 5 = 7 with no remainder ✓

Common Mistakes

Avoid these frequent errors

Adding denominators instead of finding LCM
Don't add 7 + 5 = 12! This gives the wrong denominator for fraction operations. Adding doesn't find the smallest number divisible by both. Always find the actual LCM by listing multiples or using prime factorization.

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 7 and 5 equal to 7 × 5?

Because 7 and 5 are coprime - they share no common factors except 1. When numbers have no common factors, their LCM always equals their product: $7 \times 5 = 35$ .

What's the difference between LCM and GCD?

LCM finds the smallest number both denominators divide into, while GCD finds the largest number that divides both. For coprime numbers like 7 and 5: LCM = 35, GCD = 1.

Do I always need to list all the multiples?

Not always! For coprime numbers (like 7 and 5), you can simply multiply them. For numbers with common factors, use prime factorization or the listing method.

How do I know when I've found the LCM?

The LCM is the first number that appears in both lists of multiples. In this case, 35 is the first common multiple of both 7 and 5.

Why can't I just use 70 or 105 as the common multiple?

You could use 70 or 105, but we want the least common multiple. Using 35 instead of larger multiples makes fraction calculations much easier!

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