Problem 60 - Algebra 1: Concepts and Skills

Question

List all the common factors of the pair of numbers. $$ 84,154 $$

Step-by-Step Solution

Answer

The common factors of 84 and 154 are 1, 2, 7, and 14.

Step 1: Find Factors of the First Number

Start by listing the factors of 84, which are numbers that can divide 84 without leaving a remainder. The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

Step 2: Find Factors of the Second Number

Next, find the factors of 154. The factors of 154 are: 1, 2, 7, 14, 11, 22, 77, 154.

Step 3: Identify Common Factors

Compare the two lists of factors and identify the common factors. The common factors of 84 and 154 are 1, 2, 7, and 14.

Key Concepts

Factors

When working with numbers, it's essential to understand what factors are. Factors are numbers that you can multiply together to get another number. For example, to find the factors of 84, you look for pairs of numbers that multiply to give you 84. These number pairs include (1, 84), (2, 42), (3, 28), and so on. Each of these pairs are factors because they divide 84 without leaving a remainder.

To find the factors of any number, simply test whether dividing it by consecutive whole numbers results in a whole number quotient:

• If it does, both the divisor and the quotient are factors.
• Keep testing until the divisor is greater than the square root of the number.

This method will help you list all factors, making it easier to compare when determining common factors between two numbers.

Divisibility

Divisibility plays a crucial role in identifying factors. A number is divisible by another if, after division, there is no remainder. For example, 84 is divisible by 2 because 84 divided by 2 equals 42, with a remainder of 0.

To quickly check divisibility:

• By 2: If the number ends in an even digit (0, 2, 4, 6, 8).
• By 3: If the sum of the digits is divisible by 3.
• By 5: If the number ends in 0 or 5.

Understanding divisibility shortcuts allows for faster identification of factors, easily leading to a list that helps in finding common factors between numbers like 84 and 154.

Prime Factorization

Prime factorization involves breaking down a number into the product of its prime factors. Prime factors are factors that are prime numbers, such as 2, 3, 5, and 7. Every number can be expressed as the product of prime factors, a method that simplifies finding common factors.

For instance, let's break down 84:

• Start by dividing 84 by the smallest prime number 2: 84 ÷ 2 = 42.
• Divide 42 by 2: 42 ÷ 2 = 21.
• Since 21 is odd, divide it by the next smallest prime, 3: 21 ÷ 3 = 7.
• Finally, 7 is a prime number, so the prime factors of 84 are 2, 2, 3, and 7.

Applying the same method to 154 results in the prime factors 2, 7, and 11. With these, you can easily compare and find the common prime factors, which aid in understanding the full factorization of both numbers.