Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

(a) Find the number of integers between 32 and 395 that are divisible by 6 . (b) Find their sum.

Step-by-Step Solution

Answer

There are 60 integers divisible by 6, and their sum is 18900.

Step 1: Identify the First Integer Divisible by 6

To find the first integer divisible by 6 in the range 32 to 395, we divide 32 by 6, resulting in approximately 5.33. Therefore, the smallest integer is obtained by rounding up to the nearest whole number, resulting in 6 times 6, or 36.

Step 2: Identify the Last Integer Divisible by 6

Divide 395 by 6, which gives approximately 65.83. The largest integer divisible by 6 is 6 times 65, resulting in 390.

Step 3: Calculate the Number of Integers Divisible by 6

To determine the number of integers divisible by 6 between 36 and 390, set up an arithmetic sequence. The first term is 36 and the last term is 390 with a common difference of 6. Use the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n-1)d \). Solve for \(n\) as follows: \(390 = 36 + (n-1)6\). This simplifies to \(n = 60\), so there are 60 such integers.

Step 4: Sum the Integers Divisible by 6

The sum of an arithmetic sequence can be found using the formula \( S_n = \frac{n}{2} (a_1 + a_n) \). Here \( n = 60 \), \( a_1 = 36 \), and \( a_n = 390 \). Substitute these values to get \( S_{60} = \frac{60}{2} (36 + 390) = 18900 \).

Key Concepts

Divisibility

Divisibility is a fundamental concept in arithmetic that deals with whether one integer can be divided by another without leaving a remainder. For instance, when we say a number is divisible by 6, it means that the number can be evenly divided by 6 with no remainder. To determine if a number is divisible by 6, it must meet two conditions: it should be divisible by both 2 and 3.

For example, consider the number 36. To confirm its divisibility by 6, check the following:

• It is an even number, meaning it’s divisible by 2.
• The sum of its digits (3 + 6) equals 9, which is divisible by 3.

Therefore, 36 satisfies both conditions and hence, is divisible by 6. Understanding divisibility is crucial for solving arithmetic sequence problems where determining the range of consecutive divisible integers is necessary.

Integer Range

Integer range refers to a set of whole numbers lying between two specified values. In this context, the task is to find integers divisible by a particular number within a given range. Identifying such sequences requires knowing both the smallest and largest multiples of the number in that range.

For our exercise, the range is from 32 to 395. To find integers divisible by 6 within this range, we divide the endpoints by 6 to ascertain the nearest multiples that fit the specified interval:

• The smallest integer divisible by 6 greater than 32 is 36 since 32 divided by 6 gives approximately 5.33.
• The largest integer divisible by 6 less than 395 is 390 because 395 divided by 6 equals approximately 65.83, corresponding to 6 times 65, which is 390.

Recognizing these bounds is key to solving problems involving integer ranges and divisibility.

Sum of Integers

The sum of integers refers to the total obtained by adding numbers together. Calculating this sum for a series of numbers is a common task in mathematics, often requiring specific formulas to simplify the process.

For arithmetic sequences, such as a series of numbers divisible by 6, we utilize the formula for the sum of an arithmetic sequence: \[ S_n = \frac{n}{2} (a_1 + a_n) \]Here, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term.

In our exercise:

• The number of terms \( n \) is 60, which we determined by solving the arithmetic sequence equation.
• The first and last terms are \( a_1 = 36 \) and \( a_n = 390 \) respectively.

Substituting these into our formula, we find that the sum \( S_{60} = \frac{60}{2} (36 + 390) \), which equals 18,900. This formula greatly simplifies the calculation of sums for evenly spaced integers, making complex additions more manageable.

Arithmetic Progression

Arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference. Understanding arithmetic progression is crucial when dealing with series of numbers that follow a predictable pattern.

In problems involving divisibility, like finding integers divisible by a given number, arithmetic progression helps in structuring the sequence. Here, integers divisible by 6 within a specific range form an arithmetic sequence.

The elements of this sequence are determined as follows:

• The first term, \( a_1 \), is the smallest number divisible by 6 in the range (36 in our example).
• The common difference \( d \) is 6, as we are considering numbers divisible by 6.
• The nth term formula is given by \( a_n = a_1 + (n-1)d \), leading us to find the total number of terms in the sequence.

This structured approach allows us to handle and solve problems involving arithmetic sequences efficiently, providing a clear path for calculations.