Problem 95 - Algebra 1: Concepts and Skills

Question

Decide whether the statement is true or false. If it is false, correct the statement to make it true. The greatest common factor of 10 and 30 is 5.

Step-by-Step Solution

Answer

The statement 'The greatest common factor of 10 and 30 is 5' is false. The correct statement is 'The greatest common factor of 10 and 30 is 10.'

Step 1: Understand the GCD concept

The concept of greatest common divisor (GCD) or greatest common factor (GCF) is essential in mathematics. The GCF of two numbers is the largest number that divides both of them without leaving a remainder.

Step 2: Calculate the GCD of 10 and 30

To find the GCD of 10 and 30, list all of the prime factors of each number and multiply those factors both numbers have in common. The prime factors of 10 are 2 and 5. The prime factors of 30 are 2, 3, and 5. The prime factor that 10 and 30 have in common is 2 and 5. So, multiply 2 and 5 to give 10, which is the GCD.

Step 3: Compare the calculated GCD with the one stated

The GCD calculated (10) is not the same as the one stated in the problem (5). As such, the statement is false.

Step 4: Correct the statement

To make the statement true, it should be corrected to say: 'The greatest common factor of 10 and 30 is 10.'

Key Concepts

GCD calculation

When we talk about the greatest common divisor (GCD), we are looking for the largest number that can evenly divide two or more numbers without leaving a remainder.
It is sometimes referred to as the greatest common factor (GCF). The GCD comes in handy especially in simplifying fractions or dividing quantities into smaller equal parts.
To calculate the GCD:

• List the factors of each number.
• Identify the common factors shared between the numbers.
• Select the greatest among these common factors as the GCD.

For example, let's explore the numbers 10 and 30. By listing out the factors, we determine that:

• The factors of 10 are 1, 2, 5, and 10.
• The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

The largest common factor for both numbers is 10.
Therefore, the GCD of 10 and 30 is clearly 10. This confirms that the initial statement provided was incorrect.

Prime Factorization

Prime factorization is a method used to break down a number into prime numbers that multiply together to give the original number.
This process is not only fundamental in finding the GCD, but also in understanding the composition of numbers.
To perform prime factorization:

• Continually divide the number by the smallest prime number (starting from 2) until you reach 1.
• Note down each prime factor as you divide the original number.

Looking at our problem with the numbers 10 and 30:

• The prime factors of 10 are 2 and 5. Dividing, we get: \(10 \div 2 = 5 \) and \(5 \div 5 = 1\).
• For 30, dividing gives \(30 \div 2 = 15\), \(15 \div 3 = 5\), and \(5 \div 5 = 1\), leading to prime factors 2, 3, and 5.

The common prime factors between 10 and 30 are 2 and 5.
By multiplying these common factors (\(2 \times 5\)), we confirm the GCD is indeed 10.

Mathematical Correction

Mathematical corrections often involve reassessing a solution or a statement to ensure accuracy.
Sometimes initial assumptions or calculations might be incorrect, leading to false statements.
It's essential to thoroughly verify outcomes in mathematics to avoid misinterpretation.
In our previous example, the problem initially stated the GCD of 10 and 30 as 5, which was incorrect. To correct this statement:

• Calculate both numbers' actual GCD correctly through prime factorization.
• Replace the false assertion with the accurate result.

The correction would thus declare: 'The greatest common factor of 10 and 30 is 10.'
This illustrates the importance of detailed checking and accurate calculation to maintain mathematical integrity.