Problem 2 - Elementary and Intermediate Algebra

Question

In each of Exercises \(1-4\) match the description with the appropriate number from the list on the right. A number that has 3 as a factor A. 2 B. 7 C. 60 D. 65

Step-by-Step Solution

Answer

60

Step 1: Define the problem

Identify the number that has 3 as a factor from the given choices: 2, 7, 60, 65.

Step 2: Check divisibility by 3

For a number to have 3 as a factor, it must be divisible by 3. This can be checked by dividing each number by 3.

Step 3: Test each number

Perform the following division: 1. 2 divided by 3: \(2 \div 3 = 0.666...\) (not divisible)2. 7 divided by 3: \(7 \div 3 = 2.333...\) (not divisible)3. 60 divided by 3: \(60 \div 3 = 20\) (divisible)4. 65 divided by 3: \(65 \div 3 = 21.666...\) (not divisible)

Step 4: Identify the correct number

From the division tests, 60 is the only number that is divisible by 3. Therefore, 60 is the number that has 3 as a factor.

Key Concepts

factors

When we talk about factors, we refer to numbers that can be multiplied together to get another number. For instance, if we say a number 'A' is a factor of number 'B', it means 'A' can divide 'B' without leaving any remainder. For example, 3 and 4 are factors of 12 because 3 multiplied by 4 equals 12. Identifying factors is crucial in determining divisibility and solving many mathematical problems. Knowing this concept well will help you understand the logical flow of solving divisibility-related exercises.

divisibility by 3

To determine if a number is divisible by 3, a simple rule can be used: Sum the digits of the number and check if the result is divisible by 3. If it is, then the original number is also divisible by 3. For example, take the number 60. The sum of its digits is 6 + 0 = 6. Since 6 is divisible by 3, 60 is also divisible by 3. This rule can save you time from performing actual division.

division

Division is one of the four basic arithmetic operations. It is the process of distributing a group of things into equal parts. In mathematical terms, when you divide a number 'A' by another number 'B', you are essentially finding out how many times 'B' fits into 'A'. For instance, in the step-by-step solution, you see the division operation such as 60 divided by 3. Here, 60 (dividend) is divided by 3 (divisor), and the quotient is 20, meaning 3 fits into 60 exactly 20 times. Understanding division is fundamental because it helps in breaking down complex problems into simpler ones.