Question

How many factors does the first number in a series of 3 consecutive numbers have, if that series has the sum of 57 ? A. 6 B. 5 C. 4 Answer D. 3 E. 2

Short answer

The number of factors is 6.

Step-by-step solution

Identify the series

Let the three consecutive numbers be represented as \(x, x+1, x+2\).

Sum of the series

The sum of these three numbers is given by: \[x + (x+1) + (x+2) = 57\].

Simplify the equation

Combine like terms: \[3x + 3 = 57\].

Solve for x

Subtract 3 from both sides: \[3x = 54\]. Then divide by 3: \[x = 18\]. So, the three consecutive numbers are 18, 19, and 20.

Find factors of the first number

The first number is 18. To find the factors of 18, determine the numbers that divide 18 without leaving a remainder. The factors are: 1, 2, 3, 6, 9, 18.

Count the factors

Count the number of factors of 18. There are 6 factors.

Key concepts

consecutive numbers

Consecutive numbers are numbers that follow each other in order. They increase by exactly 1 from one number to the next. For example, in the series 18, 19, and 20, each number is one more than the previous one. When we deal with problems involving consecutive numbers, we often represent them algebraically. For example, if the first number is denoted by x, the next two numbers would be x+1 and x+2.
Understanding consecutive numbers is crucial because it allows us to set up simple algebraic equations to find unknown values. In our problem, the three consecutive numbers are represented as x, x+1, and x+2. This representation helps us to easily work with the sum and find the individual numbers.

sum of series

The sum of a series of numbers is simply the total when you add all the numbers together. In our exercise, we need to find the sum of three consecutive numbers. We know the sum is 57. Using our algebraic representation, we write the sum as \[x + (x+1) + (x+2) = 57\]
Combining like terms in this equation simplifies it to much easier terms: \[3x + 3 = 57\]
A key point about series sums is that it translates word problems into mathematical equations that we can solve step-by-step. Breaking down and simplifying the series sum helps us solve for the unknown variable.

solving algebraic equations

Algebraic equations can sometimes look tricky, but with a series of steps, we can solve them easily. Let’s go through the process for our equation: \[3x + 3 = 57\]
First, subtract 3 from both sides to isolate the term with the variable x:\[3x = 54\]
Next, divide both sides by 3 to solve for x:\[x = 18\]
By following these steps, we find that x equals 18. This means our original series of consecutive numbers is 18, 19, and 20. Understanding how to break down and solve algebraic equations is a powerful tool for many types of math problems.

finding factors

Finding the factors of a number means determining all the numbers that can divide into it without leaving a remainder. For the number 18, we need to find which numbers divide 18 exactly. These factors are: 1, 2, 3, 6, 9, 18.
To make sure we have all factors, we start with 1 and go up to the number itself, checking divisibility. Listing the factors of 18, we count them and find there are 6 total. This counting tells us how many factors the number has. Understanding factors is useful not just for solving homework problems, but for more complex mathematical tasks such as finding greatest common divisors or simplifying fractions.