CAT Divisibility rules of numbers which are products of two or more coprime numbers and problems based on it. - Practice Questions & MCQ

In this concept, we will explore the divisibility rules specifically for numbers that are products of two or more coprime numbers.

Definition:

Coprime numbers, also known as relatively prime or mutually prime numbers, are a set of numbers that have no common factors except 1. When two or more coprime numbers are multiplied together, the resulting number inherits certain divisibility properties.

Divisibility Rules:

In this section, we will discuss the divisibility rules for numbers that are products of two or more coprime numbers:

Rule 1: If a number is divisible by each of the coprime numbers which are being multiplied together, it is also divisible by their product.

Rule 2: When checking divisibility, only consider the factors that are common to all the coprime numbers being multiplied. Ignore any other factors.

Rule 3: If the product of two or more coprime numbers is divisible by a particular number, then each of the coprime numbers individually is also divisible by that number.

Example Problems:

Let's solve a few example problems to better understand the application of the divisibility rules:

Example 1:

Question: Determine if the number 4536 is divisible by 21.

Solution:

Since 21 is the product of two coprime numbers, 3 and 7, we can apply Rule 1. First, let's check the divisibility of 4536 by 3:

We know that a number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits in 4536 is 4 + 5 + 3 + 6 = 18, which is divisible by 3. Therefore, 4536 is divisible by 3.

Now let's check the divisibility of 4536 by 7:

While there is no specific divisibility rule for 7, we can reduce 4536 to its alternate sum of groups of three digits (45 - 36 = 9). Since 9 is divisible by 7, 4536 is also divisible by 7.

Therefore, 4536 is divisible by both 3 and 7 i.e. it will be divisible by 21.

Example 2:

Question: Find the number of positive integers less than 1000 that are divisible by both 4 and 5.

Solution:

Since 4 and 5 are coprime numbers, we can apply Rule 2 here. To find the number of positive integers divisible by both 4 and 5, we need to find their least common multiple (LCM).

The LCM of 4 and 5 is 20. Since 1000 is divisible by 20, we divide 1000 by 20 to find the number of multiples:

1000 ÷ 20 = 50

Therefore, there are 50 positive integers less than 1000 that are divisible by both 4 and 5.

Tips and Tricks:

1. Remember the basic divisibility rules for prime numbers, as they are often used in determining the divisibility of products of coprime numbers.
2. To simplify calculations, break down large numbers into their prime factors and apply the divisibility rules separately for each prime factor.
3. Practise previous year's management entrance questions to gain a better understanding of how the concept of divisibility is applied in problem-solving.
4. Use the concept of prime factorization to determine which prime factors are relevant for checking divisibility.

By mastering the divisibility rules of numbers that are products of coprime numbers, you can enhance your problem-solving skills and improve your performance in management entrance exams.