Problem 69 Find the GCF. $$25,125,625$$... [FREE SOLUTION]

Chapter 2: Problem 69

Find the GCF. $$25,125,625$$

Short Answer

Expert verified

The Greatest Common Factor (GCF) of 25, 125, and 625 is 25.

Step by step solution

Prime Factorization

Perform prime factorization on each number. For 25, the prime factors are \[25=5*5\]. For 125, the prime factors are \[125=5*5*5\]. For 625, the prime factors are \[625=5*5*5*5\].

Identify the Common Factors

Evaluate the prime factorization to pinpoint the common factors present in all three numbers. In this case, the common factors of 25, 125, and 625 are \[5*5\].

Compute the GCF

The GCF is obtained by multiplying these common prime factors. So, \[GCF = 5*5\]

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Key Concepts

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

Prime Factorization

Prime factorization is like breaking down a number into a product of prime numbers. Think of it as finding the building blocks that make up the number. Every number can be represented as a combination of primes. For example:

The number 25 can be expressed as $25 = 5 \times 5$ because 5 is a prime number.
The number 125 breaks down to $125 = 5 \times 5 \times 5$.
The number 625 can be factored as $625 = 5 \times 5 \times 5 \times 5$.

This step is crucial when trying to find the greatest common factor (GCF) because it allows us to see the basic components of each number and identify which primes they have in common.

Common Factors

Once you have the prime factorizations, the next step is to look for common factors. Common factors are numbers that are shared between two or more lists of factors. In this case:

For 25, the factors are $5 \times 5$.
For 125, the factors are $5 \times 5 \times 5$.
For 625, the factors are $5 \times 5 \times 5 \times 5$.

We can find that each of these numbers shares the factor $5 \times 5$. This means $5 \times 5$ appears in the factorization of all three numbers. Identifying these shared primes is key to determining the GCF because only the common factors will contribute to it.

Multiplying Factors

Multiplying factors is the final step in figuring out the GCF. Once we have determined the common factors, we simply multiply them to find the greatest common factor. For our numbers, 25, 125, and 625, the common factors are $5 \times 5$. Therefore, we calculate the GCF as follows:

Multiply the common prime factors: $5 \times 5 = 25$.

By multiplying these common factors together, we find that the GCF of 25, 125, and 625 is 25. This process shows that despite how large the individual numbers are, their greatest common factor can be found by focusing only on what they share after performing prime factorization.

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91影视

Find the GCF. $$25,125,625$$

Short Answer

Step by step solution

Prime Factorization

Identify the Common Factors

Compute the GCF

Key Concepts

Prime Factorization

Common Factors

Multiplying Factors

One App. One Place for Learning.

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