LCM Exercises: Step-by-Step Solutions

by TextBrain Team 38 views

Hey guys! Let's break down how to find the Least Common Multiple (LCM) for these sets of numbers. The LCM is super useful in math, especially when you're trying to add or subtract fractions with different denominators. Basically, it's the smallest number that all the numbers in the set can divide into evenly. Let's jump right in!

1) Finding the LCM of 70, 30, and 45

To find the Least Common Multiple (LCM) of 70, 30, and 45, we'll use the prime factorization method. This involves breaking down each number into its prime factors, which are the prime numbers that multiply together to give the original number. Let's start by finding the prime factorization of each number:

  • Prime factorization of 70:

    70 = 2 x 35 = 2 x 5 x 7. So, the prime factors of 70 are 2, 5, and 7.

  • Prime factorization of 30:

    30 = 2 x 15 = 2 x 3 x 5. Thus, the prime factors of 30 are 2, 3, and 5.

  • Prime factorization of 45:

    45 = 3 x 15 = 3 x 3 x 5 = 32 x 5. Therefore, the prime factors of 45 are 3 and 5.

Now that we have the prime factorizations, we'll identify the highest power of each prime factor that appears in any of the factorizations. This will form the basis for calculating the LCM. Here's what we have:

  • The highest power of 2 is 21 (from 70 and 30).
  • The highest power of 3 is 32 (from 45).
  • The highest power of 5 is 51 (from 70, 30, and 45).
  • The highest power of 7 is 71 (from 70).

To find the LCM, we multiply these highest powers of prime factors together:

LCM (70, 30, 45) = 21 x 32 x 51 x 71 = 2 x 9 x 5 x 7 = 630.

So, the Least Common Multiple of 70, 30, and 45 is 630. This means that 630 is the smallest number that is divisible by all three numbers.

In summary:

  • 70 = 2 x 5 x 7
  • 30 = 2 x 3 x 5
  • 45 = 32 x 5
  • LCM (70, 30, 45) = 2 x 32 x 5 x 7 = 630

2) Calculating the LCM of 100, 25, and 60

Alright, let's tackle the next one! We need to find the LCM of 100, 25, and 60. Again, we'll use the prime factorization method to break each number down into its prime factors. Here's how it goes:

  • Prime factorization of 100:

    100 = 2 x 50 = 2 x 2 x 25 = 22 x 52. So, the prime factors of 100 are 2 and 5.

  • Prime factorization of 25:

    25 = 5 x 5 = 52. Thus, the prime factor of 25 is 5.

  • Prime factorization of 60:

    60 = 2 x 30 = 2 x 2 x 15 = 22 x 3 x 5. Therefore, the prime factors of 60 are 2, 3, and 5.

Now that we have the prime factorizations, we'll identify the highest power of each prime factor that appears in any of the factorizations. This will form the basis for calculating the LCM. Here's what we have:

  • The highest power of 2 is 22 (from 100 and 60).
  • The highest power of 3 is 31 (from 60).
  • The highest power of 5 is 52 (from 100 and 25).

To find the LCM, we multiply these highest powers of prime factors together:

LCM (100, 25, 60) = 22 x 31 x 52 = 4 x 3 x 25 = 300.

So, the Least Common Multiple of 100, 25, and 60 is 300. This means that 300 is the smallest number that is divisible by all three numbers.

In summary:

  • 100 = 22 x 52
  • 25 = 52
  • 60 = 22 x 3 x 5
  • LCM (100, 25, 60) = 22 x 3 x 52 = 300

3) Determining the LCM of 48, 56, and 49

Next up, let's find the LCM of 48, 56, and 49. We're sticking with the prime factorization method because it’s reliable and helps us understand what's going on. Here’s the breakdown:

  • Prime factorization of 48:

    48 = 2 x 24 = 2 x 2 x 12 = 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 3 = 24 x 3. Thus, the prime factors of 48 are 2 and 3.

  • Prime factorization of 56:

    56 = 2 x 28 = 2 x 2 x 14 = 2 x 2 x 2 x 7 = 23 x 7. So, the prime factors of 56 are 2 and 7.

  • Prime factorization of 49:

    49 = 7 x 7 = 72. Therefore, the prime factor of 49 is 7.

After breaking down each number into its prime factors, we need to identify the highest power of each prime factor present in the factorizations. Let's see what we've got:

  • The highest power of 2 is 24 (from 48).
  • The highest power of 3 is 31 (from 48).
  • The highest power of 7 is 72 (from 49).

To find the LCM, we multiply these highest powers of prime factors together:

LCM (48, 56, 49) = 24 x 31 x 72 = 16 x 3 x 49 = 2352.

So, the Least Common Multiple of 48, 56, and 49 is 2352. That's the smallest number that all three of these numbers can divide into evenly.

In summary:

  • 48 = 24 x 3
  • 56 = 23 x 7
  • 49 = 72
  • LCM (48, 56, 49) = 24 x 3 x 72 = 2352

4) Finding the Least Common Multiple of 8, 14, and 20

Okay, let's keep the ball rolling! This time, we're finding the LCM of 8, 14, and 20. As always, we'll start with the prime factorization of each number:

  • Prime factorization of 8:

    8 = 2 x 4 = 2 x 2 x 2 = 23. So, the prime factor of 8 is 2.

  • Prime factorization of 14:

    14 = 2 x 7. Thus, the prime factors of 14 are 2 and 7.

  • Prime factorization of 20:

    20 = 2 x 10 = 2 x 2 x 5 = 22 x 5. Therefore, the prime factors of 20 are 2 and 5.

Now, let's identify the highest power of each prime factor from these factorizations. This will help us calculate the LCM.

  • The highest power of 2 is 23 (from 8).
  • The highest power of 5 is 51 (from 20).
  • The highest power of 7 is 71 (from 14).

To find the LCM, we multiply these highest powers of prime factors together:

LCM (8, 14, 20) = 23 x 51 x 71 = 8 x 5 x 7 = 280.

So, the Least Common Multiple of 8, 14, and 20 is 280. This is the smallest number that is divisible by all three given numbers.

In summary:

  • 8 = 23
  • 14 = 2 x 7
  • 20 = 22 x 5
  • LCM (8, 14, 20) = 23 x 5 x 7 = 280

5) Calculating the LCM of 25, 35, and 75

Last but not least, let's find the LCM of 25, 35, and 75. You know the drill by now – we'll start with prime factorization:

  • Prime factorization of 25:

    25 = 5 x 5 = 52. So, the prime factor of 25 is 5.

  • Prime factorization of 35:

    35 = 5 x 7. Thus, the prime factors of 35 are 5 and 7.

  • Prime factorization of 75:

    75 = 3 x 25 = 3 x 5 x 5 = 3 x 52. Therefore, the prime factors of 75 are 3 and 5.

Now, let's identify the highest power of each prime factor from these factorizations:

  • The highest power of 3 is 31 (from 75).
  • The highest power of 5 is 52 (from 25 and 75).
  • The highest power of 7 is 71 (from 35).

To find the LCM, we multiply these highest powers of prime factors together:

LCM (25, 35, 75) = 31 x 52 x 71 = 3 x 25 x 7 = 525.

So, the Least Common Multiple of 25, 35, and 75 is 525. This is the smallest number that all three numbers divide into without any remainder.

In summary:

  • 25 = 52
  • 35 = 5 x 7
  • 75 = 3 x 52
  • LCM (25, 35, 75) = 3 x 52 x 7 = 525

Conclusion

And there you have it! We've walked through how to find the LCM for five different sets of numbers. Remember, the key is to break down each number into its prime factors and then multiply the highest powers of each prime factor together. Hope this helps you nail those LCM problems! Keep practicing, and you'll become a pro in no time!