Question

How many multiples of 3 are there among the integers 15 through 105 inclusive?

Short answer

There are 31 multiples of 3.

Step-by-step solution

- Find the First Multiple of 3

Identify the first multiple of 3 in the given range. Start at 15 and count up: 15 is a multiple of 3 since 15 ÷ 3 = 5.

- Find the Last Multiple of 3

Identify the last multiple of 3 in the given range. Start at 105 and count down: 105 is a multiple of 3 since 105 ÷ 3 = 35.

- Use the Formula for Arithmetic Sequences

The multiples of 3 form an arithmetic sequence with the first term (a) being 15 and the common difference (d) being 3. Use the formula for the nth term of an arithmetic sequence: \[ a_n = a + (n-1)d \] Set \( a_n \) to 105 and solve for n: \[ 105 = 15 + (n-1) \times 3 \]

- Solve for n

Rearrange and solve the equation: \[ 105 = 15 + 3(n-1) ewline 105 = 15 + 3n - 3 ewline 105 = 3n + 12 ewline 105 - 12 = 3n ewline 93 = 3n ewline n = 31 \] There are 31 multiples of 3 between 15 and 105 inclusive.

Key concepts

Understanding Multiples

Multiples are the result of multiplying a number by an integer. For example, the multiples of 3 are numbers like 3, 6, 9, 12, and so on. They are evenly spaced because the difference between consecutive multiples is always the original number (3 in this case). Identifying multiples within a range involves checking if numbers within that range can be expressed as the product of the original number and some integer.
In our problem, we need to find multiples of 3 in the range from 15 to 105.

Utilizing the n-term Formula in Arithmetic Sequences

The n-term formula of an arithmetic sequence helps us find any term in the sequence if we know the first term and the common difference. The nth term formula is represented as: \( a_n = a + (n-1)d \)
Here, \( a \) is the first term, \( d \) is the common difference (the constant difference between consecutive terms), and \( n \) is the term number we are trying to find. This is immensely useful for finding the number of terms and specific terms in a sequence.
In our exercise, we set up our formula with the first term (15), the common difference (3), and the last term (105). Solving for \( n \) tells us how many terms are in our sequence.

Grasping Integer Sequences

An integer sequence is a list of numbers in which each member of the list is an integer. When these integers are arranged in a particular order and follow a specific rule, they form a sequence. For example, an arithmetic sequence like the multiples of 3 in our range (15, 18, 21, ... , 105) increment by 3 with each subsequent term.
Understanding these sequences is crucial for identifying patterns and solving related mathematical problems efficiently.

Concept of Inclusive Range

An inclusive range in mathematics means that the range includes the endpoints. For instance, in the range from 15 to 105 inclusive, both 15 and 105 are part of the range.
In our exercise, we make sure to include both 15 and 105 when counting the multiples of 3. This inclusive nature ensures our counting is accurate and all terms within that boundary are considered effectively.