Question

Find the number of positive integers up to 100 which are not divisible by any 2,3 and 5 ? (a) 24 (b) 25 (c) 26 (d) 27

Short answer

Answer: There are 30 positive integers up to 100 which are not divisible by any of the given primes.

Step-by-step solution

Count Individual Divisible Integers

Count the positive integers up to 100 which are divisible by 2, 3, and 5 individually. Divisible by 2: There are 50 numbers (2, 4, 6, ..., 100) Divisible by 3: There are 33 numbers (3, 6, 9, ..., 99) Divisible by 5: There are 20 numbers (5, 10, 15, ..., 100)

Count Divisible Integers for Prime Combinations

Now we need to count the integers which are divisible by the combinations of given primes (2*3, 2*5, 3*5). Divisible by 2*3 (6): There are 16 numbers (6, 12, 18, ..., 96) Divisible by 2*5 (10): There are 10 numbers (10, 20, 30, ..., 100) Divisible by 3*5 (15): There are 6 numbers (15, 30, 45, ..., 90)

Count Divisible Integers for All Prime Combinations

Now we need to count the integers which are divisible by the combination of all given primes (2*3*5). Divisible by 2*3*5 (30): There are 3 numbers (30, 60, 90)

Apply Inclusion-Exclusion Principle

Now we will use the Inclusion-Exclusion Principle to find out how many numbers are divisible by these primes individually and their combinations. Total numbers = Numbers divisible by 2 + Numbers divisible by 3 + Numbers divisible by 5 - (Numbers divisible by 2*3 + Numbers divisible by 2*5 + Numbers divisible by 3*5) + Numbers divisible by 2*3*5 Total numbers = 50 + 33 + 20 - (16 + 10 + 6) + 3 = 103 - 32 = 71

Find the Answer

Now we have a total of 71 numbers which are divisible by at least one of the given primes or their combinations. Since we need to find the numbers which are not divisible, we'll subtract this total from the total number of positive integers up to 100. Answer = Total positive integers up to 100 - Total numbers divisible by primes Answer = 100 - 71 = 29 However, we need to consider the fact that we started counting from 2, which means we've left out the number 1. Since 1 is not divisible by any of 2, 3, or 5, we need to add 1 to our answer. Final Answer = 29 + 1 = 30 So, there are 30 positive integers up to 100 which are not divisible by any of 2, 3, and 5. None of the given options matches the correct answer.

Key concepts

Divisibility Rules

Divisibility rules are shortcuts that help us to quickly determine if one number is a divisor of another without having to do the full division. For instance, a number is divisible by 2 if its last digit is even, and it is divisible by 3 if the sum of its digits is a multiple of 3. Divisibility by 5 requires the number to end in a 0 or a 5. These rules are crucial when identifying patterns in numbers, especially in the context of finding quantities of numbers with specific divisibility properties, like in the exercise where we are concerned with numbers up to 100 and their divisibility by 2, 3, or 5.

Understandably, mastering these rules enhances one's speed and accuracy in quantitative aptitude exercises. In our original problem, we applied these divisibility rules to count how many numbers up to 100 are divisible by 2, 3, and 5. This provided a foundational step in applying the Inclusion-Exclusion Principle to identify the numbers not divisible by these values, by systematically including and excluding counts based on multiple divisibilities.

Prime Numbers

Prime numbers are the building blocks of the number system. They are defined as positive integers greater than 1 that have no positive integer divisors other than 1 and themselves. The first few prime numbers are 2, 3, 5, 7, 11, and so on. In our problem, we focus on 2, 3, and 5, which are prime numbers and important because they represent the prime factors of many other numbers.

Understanding primes is critical when solving problems that involve divisibility, as primes can only be divided evenly by 1 or the number itself. This property makes them vital for determining divisibility by combinations of primes, which is often necessary for the Inclusion-Exclusion Principle in quantitative exercises.

Positive Integers

Positive integers are all the whole numbers greater than zero that we use regularly for counting and ordering. They are an essential concept in mathematics, forming the basis for most arithmetic operations and number theory. In the context of the given exercise, we considered the set of positive integers from 1 to 100. These numbers are the universe we analyze to find those not divisible by any of the given primes.

When dealing with a large set of positive integers, tactics like the Inclusion-Exclusion Principle become necessary to track and manage the subsets of numbers that meet specific criteria, such as being divisible by 2, 3, and 5. This helps simplify complex problems and is a technique often applied in combinatorics and probability.

Quantitative Aptitude

Quantitative aptitude is a measure of a person's ability to handle numbers and perform mathematical operations. It's often tested in competitive exams and is vital for anyone involved in analytics or professions that require strong problem-solving skills. This skillset includes understanding numerical sequences, applying mathematical concepts like the Inclusion-Exclusion Principle, and determining patterns in numeracy.

In our exercise, quantitative aptitude was central in identifying the correct use of the Inclusion-Exclusion Principle. This applies not only to solving the problem but also to understanding that none of the answer choices matched the correct answer, highlighting another critical ability in quantitative aptitude: error detection and problem solving beyond the apparent answer choices.