Problem 37 - Algebra for JEE ADVANCED

Question

How many numbers can be made with the digits \(3,4,5,6,7,8\) lying between 3000 and 4000, which are divisible by 5 while repetition of any digit is not allowed in any number? a. 60 b. 12 c. 120 d. 24

Step-by-Step Solution

Answer

b. 12

Step 1: Understanding the Criteria

We need to form a number using the provided digits that lies between 3000 and 4000 and is divisible by 5. Additionally, we cannot repeat any digit.

Step 2: Determining the Thousands Place

To ensure the number is between 3000 and 4000, the thousands digit must be 3. Thus, the number will be in the form of \(3xyz\), where \(x\), \(y\), and \(z\) are other digits selected from the remaining digits.

Step 3: Ensuring Divisibility by 5

For a number to be divisible by 5, the units digit (\(z\)) must be either 0 or 5. Since our available digits are 3, 4, 5, 6, 7, and 8, the units digit must be 5.

Step 4: Constructing the Number

With the thousands place being 3 and the units place being 5, the number takes the form \(3xy5\). We need to determine the suitable values for \(x\) and \(y\) using the digits 4, 6, 7, and 8.

Step 5: Calculating Combinations for Hundreds and Tens

There are 4 digits (4, 6, 7, and 8) left for digits \(x\) and \(y\). Choose any 2 digits for \(x\) in 4 ways, and the remaining digit for \(y\) in 3 ways. Thus, the total numbers can be calculated as \(4 \times 3 = 12\).

Step 6: Final Calculation and Verification

Verify whether the selection satisfies all the conditions: 12 numbers are formed which satisfy that they lie between 3000 and 4000 and are divisible by 5 without repeating any digit.

Key Concepts

Number Divisibility

Divisibility is a concept in mathematics where one number can be divided by another without leaving a remainder. In our exercise, we are particularly concerned with divisibility by 5.
For a number to be divisible by 5, its last digit must be either 0 or 5.
This is a key rule derived from the basic fact that any multiple of 5 will end with these digits.

• In this problem, the digits we can use are 3, 4, 5, 6, 7, and 8.
• Since these do not contain 0, the digit 5 is our only choice to ensure the number meets the divisibility rule.

Therefore, the task is naturally limited in terms of our selection for the units place, highlighting the foundational significance of understanding divisibility rules when tackling such exercises.

Digit Arrangement

Arranging digits to form numbers is central to combinatorics. It involves choosing which digits appear in which positions to fulfill certain criteria.
In this exercise, we arrange the digits 3, 4, 5, 6, 7, and 8.
Let's break down the arrangement process:

• The thousands place must be 3 to satisfy the range requirement.
• The units place must be 5 to ensure divisibility by 5, as discussed earlier.
• Once these positions are fixed, the remaining digits (4, 6, 7, 8) are available for the hundreds and tens places.

For the two remaining places, there are permutations of the remaining four digits. Begin by selecting a digit for the hundreds place; this decision affects the tens place, which must be chosen from the remaining unused digits.
This method of sequentially filling digit positions ensures each number fits the stated requirements.

Constraints in Number Formation

Constraints are the rules that restrict the choice of digit in specific places when forming numbers. In this problem, constraints play a significant part in guiding the formation process.
Here's how constraints influence our solution:

• The number must be between 3000 and 4000. This means the thousands digit is fixed as 3.
• The divisibility by 5 condition determines the units digit to be 5.
• Since no digit can be repeated and only specific digits are available, every choice for hundreds or tens affects subsequent selections.

By understanding these constraints, we efficiently reduce our potential number combinations.
This problem reduces to arranging the four remaining digits after fixing the thousands and units digits. Adhering to these constraints leads to a controlled and structured approach in number formation, allowing us to precisely determine that there are 12 viable combinations fitting all criteria.