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Chapter 8: Problem 10

Find the number of positive integers not exceeding 100 that are not divisible by 5 or by 7 .

Short Answer

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68

Step by step solution

01

Identify Total Numbers

The first step is to identify the total number of positive integers not exceeding 100. This is simply 100, as we are considering the numbers from 1 to 100.
02

Count Numbers Divisible by 5

Next, count the numbers that are divisible by 5. The sequence of numbers divisible by 5 up to 100 is 5, 10, 15, ..., 100. This forms an arithmetic sequence with the first term (a) as 5 and the common difference (d) as 5. The number of terms (n) in this sequence can be found using the formula for the nth term of an arithmetic sequence: \[ an = a + (n-1)d \] Set the nth term (an) to 100 and solve for n: \[ 100 = 5 + (n-1) \times 5 \] \[ 100 = 5 + 5n - 5 \] \[ 100 = 5n \] \[ n = 20 \] There are 20 numbers divisible by 5.
03

Count Numbers Divisible by 7

Count the numbers that are divisible by 7 up to 100. The sequence of numbers divisible by 7 is 7, 14, 21, ..., 98. This forms an arithmetic sequence with the first term (a) as 7 and the common difference (d) as 7. Using the nth term formula: \[ an = a + (n-1)d \] Set the nth term (an) to the largest number not exceeding 100 and solve for n: \[ 98 = 7 + (n-1) \times 7 \] \[ 98 = 7 + 7n - 7 \] \[ 98 = 7n \] \[ n = 14 \] There are 14 numbers divisible by 7.
04

Count Numbers Divisible by Both 5 and 7

Find the numbers that are divisible by both 5 and 7. Any such number must be divisible by their least common multiple (LCM). The LCM of 5 and 7 is 35. The sequence of numbers up to 100 that are divisible by 35 is 35, 70. This forms an arithmetic sequence with the first term (a) as 35 and the common difference (d) as 35: \[ an = a + (n-1)d \] Set the nth term (an) to the largest number not exceeding 100 and solve for n: \[ 70 = 35 + (n-1) \times 35 \] \[ 70 = 35 + 35n - 35 \] \[ 70 = 35n \] \[ n = 2 \] There are 2 numbers divisible by both 5 and 7.
05

Apply Inclusion-Exclusion Principle

Use the Inclusion-Exclusion Principle to find the total number of positive integers not divisible by 5 or by 7. Count the numbers divisible by 5 (20) and 7 (14), subtract the numbers divisible by both (2): \[ 20 + 14 - 2 = 32 \] There are 32 numbers divisible by 5 or by 7.
06

Calculate the Result

Subtract the number of integers divisible by 5 or 7 from the total integers not exceeding 100 to get the result: \[ 100 - 32 = 68 \] Thus, there are 68 numbers not exceeding 100 that are not divisible by 5 or by 7.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inclusion-Exclusion Principle
The Inclusion-Exclusion Principle is a fundamental concept in combinatorics. It helps us count the number of elements in the union of multiple sets by including the sizes of individual sets and excluding the sizes of overlapping sets.
In this problem, we want to find numbers up to 100 that are not divisible by 5 or 7. Using the Inclusion-Exclusion Principle, we first count the numbers divisible by 5 and those divisible by 7.
Then, we subtract the count of numbers divisible by both to avoid double-counting.
  • Count divisible by 5: 20
  • Count divisible by 7: 14
  • Count divisible by both 5 and 7: 2
Applying the principle:
\[ 20 + 14 - 2 = 32 \] Subtract this from the total (100):
\[ 100 - 32 = 68 \] We get 68 numbers that are not divisible by either.
Arithmetic Sequence
An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms.
For example, if we want to count numbers divisible by 5 up to 100, these form an arithmetic sequence: 5, 10, 15, ..., 100. Here, the first term (a) is 5, and the common difference (d) is also 5.
To find the number of such terms, we use the formula for the nth term:
\[ a_n = a + (n-1)d \] Setting the nth term (a_n) to 100 and solving for n:
\[ 100 = 5 + (n-1) \times 5 \] \[ 95 = 5n \] \[ n = 20 \] So, there are 20 terms divisible by 5. Similarly, we count numbers divisible by 7 and those divisible by 35, which are also arithmetic sequences.
Divisibility
Divisibility is a key concept in number theory and arithmetic. A number is divisible by another if it can be divided without leaving a remainder.
In our problem, we identify numbers up to 100 which are divisible by 5 or 7. By understanding divisibility, we can list numbers that meet these criteria.
  • Numbers divisible by 5: Divisibility rules tell us that if a number ends in 0 or 5, it is divisible by 5.
  • Numbers divisible by 7: This is less straightforward and often involves checking individual terms.
To see if a number is divisible by both 5 and 7, we find its Least Common Multiple (LCM). The LCM of 5 and 7 is 35. So, numbers like 35 and 70 are divisible by both.
These concepts help break down the task of counting specific groups of numbers efficiently.

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