Problem 87 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of positive numbers less than 1000 and divisible by 5 (no digit being repeated) is (A) 150 (B) 154 (C) 166 (D) None of these

Step-by-Step Solution

Answer

The correct answer is (B) 154.

Step 1: Understanding the Problem

We need to find out how many numbers less than 1000 are divisible by 5. Additionally, none of the digits in these numbers should be repeated.

Step 2: Setting Conditions

A number is divisible by 5 if its last digit is either 0 or 5. Since we are dealing with numbers less than 1000, we are considering 1-digit, 2-digit, and 3-digit numbers.

Step 3: Case 1 - 1-Digit Numbers Divisible by 5

In this case, the numbers are 0 and 5. However, since 0 is not a positive number, only 5 is considered. Thus, there is **1** number: 5.

Step 4: Case 2 - 2-Digit Numbers Divisible by 5

For these numbers, the last digit is either 0 or 5. We explore each option: **Last digit 0:** Choose a non-repeating first digit from the digits 1-9, giving us 9 possibilities (10, 20, ..., 90). **Last digit 5:** Choose a non-repeating first digit from the digits 1-9, excluding 5, giving us 8 possibilities (15, 25, ..., 95). So, there are **9 + 8 = 17** options.

Step 5: Case 3 - 3-Digit Numbers Divisible by 5

Here, three-digit combinations must be considered with the last digit as 0 or 5: **Last digit 0:** - Choose a first digit from 1 to 9 (9 options). - Choose a second digit different from the first (8 options). This gives us **9 x 8 = 72** options. **Last digit 5:** - Choose a first digit from 1 to 9, excluding 5 (8 options). - Choose a second digit different from the first and 5 (8 options). This gives us **8 x 8 = 64** options. Thus, there are **72 + 64 = 136** options.

Step 6: Adding All Cases

Combine the numbers obtained from each case: 1-digit, 2-digit, and 3-digit numbers: **1 + 17 + 136 = 154.**

Step 7: Conclusion

Based on calculation, the total number of positive numbers below 1000 and divisible by 5, with no repeated digits, is 154, which corresponds to option (B).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting and arranging objects. It helps us find the number of ways to arrange or select items in a specific manner. In the context of this exercise, combinatorics enables us to calculate how many unique numbers we can form under certain conditions.
To solve the problem of finding 3-digit numbers divisible by 5 with unique digits, we used a simple combinatorial approach. We determined the number of possible choices for different positions in a number. This involved selecting digits for each place in the number such that each digit was different from the others. This methodical process helped to ensure no repetition of digits while covering all potential ways to form numbers under the specified rules. By organizing numbers based on the divisibility rule of 5 and arranging them with unique digits, combinatorics made the counting process efficient and straightforward.

Number Theory

Number theory is a field of mathematics that studies numbers and their properties, especially integers. It plays a crucial role in understanding how numbers behave under different mathematical operations, such as divisibility. In this exercise, we applied the concept of divisibility in number theory to determine which numbers are divisible by 5.
The rule of divisibility by 5 states that a number is divisible by 5 if its last digit is either 0 or 5. We used this concept to narrow down our choices for forming numbers, checking which combinations of digits satisfied this condition without repeating digits. Number theory provides the foundational rules that guide us in determining such properties, ensuring the numbers constructed meet all the criteria posed by the problem.

3-Digit Numbers

Three-digit numbers are numbers that range from 100 to 999. They consist of a combination of hundreds, tens, and units digits. In this problem, we considered 3-digit numbers while ensuring they were also divisible by 5 and had unique digits. This means carefully selecting the hundreds, tens, and units places.
The hundreds digit must be any non-zero number from 1 to 9, the tens digit could be any number that is not the same as the hundreds digit or 5, and we selected the units place as either 0 or 5 depending on our condition for divisibility by 5. Understanding the structure of 3-digit numbers helps us efficiently organize our approach to forming numbers according to the problem's requirements.

Unique Digits

Using unique digits means each digit in a number cannot be repeated. This constraint adds an interesting twist to finding numbers as it limits the available choices for each digit. The concept of unique digits ensures that numbers are not only divisible by a certain number but also composed of naturally varied elements.
For this problem, specifically, we needed numbers with no repeating digits to meet the criterion of uniqueness. This required us to methodically choose digits such that once a digit is used, it could not be selected again for the same number, ensuring each number's uniqueness. This condition impacted the calculation as it reduced the pool of available choices and needed careful consideration in each step of forming the numbers.