Problem 13 Find the LCM. 10 and 25... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 4: Problem 13

Find the LCM. 10 and 25

Short Answer

Expert verified

LCM = 50

Step by step solution

List the Prime Factors

Find the prime factors of each number.\[10 = 2 \times 5\] \[25 = 5 \times 5\]

Identify the Highest Powers of Each Prime

In the factorizations from Step 1, identify the highest power of each prime number: \[2: 2^1\] \[5: 5^2\]

Multiply the Highest Powers

Multiply the highest powers of all primes together to get the LCM: \[LCM = 2^1 \times 5^2 = 2 \times 25 = 50\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factors

Understanding prime factors is crucial in finding the Least Common Multiple (LCM). Prime factors are the building blocks of any number. A prime number is a number greater than one that only has two divisors, 1 and itself. To find the prime factors of a number, we keep dividing by prime numbers until we are left with 1. For instance, for 10, we start with the smallest prime number, which is 2.

10 divided by 2 is 5
Next, 5 is itself a prime number

So, the prime factors of 10 are 2 and 5, which we'll write as 10 = 2 脳 5.Now let's consider 25. Since 25 is not even, we bypass 2 and try the next smallest prime, which is 5.

25 divided by 5 is 5
And 5 divided by 5 is 1

Thus, the prime factors of 25 are 5 and 5, i.e., 25 = 5 脳 5.

Highest Powers

Once we have the prime factors, the next step is finding the highest power of each prime number from the factorizations. This step ensures that the LCM covers all contributing prime factors in their maximum required amounts. From the previous section, the prime factors needed for 10 are 2 and 5, and for 25, they are 5 and 5.For number 10:

Prime factor 2 is raised to the power 1 (written as 2^1)
Prime factor 5 is also raised to the power 1 (written as 5^1)

For number 25:

Prime factor 5 is raised to the power 2 (written as 5^2)

Considering the highest powers, we take the largest power of each prime. Here, we need 2^1 and 5^2, ensuring that all factors needed for 10 and 25 are included.

Multiplication

The final step in finding the LCM is the multiplication of the highest powers of all prime numbers identified. This results in the smallest number that both original numbers perfectly divide. For our example:

We multiply 2^1 by 5^2 (which we found as the highest powers)
This gives us: LCM = 2 脳 25 = 50

So, the LCM of 10 and 25 is 50. Remember, this method ensures that our LCM accounts for all prime factors in their necessary counts. Thus, our solution is complete and correct by combining and multiplying the highest powers of their prime factors together.

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91影视

Find the LCM. 10 and 25

Short Answer

Step by step solution

List the Prime Factors

Identify the Highest Powers of Each Prime

Multiply the Highest Powers

Key Concepts

Prime Factors

Highest Powers

Multiplication

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