Finding the Greatest 4-Digit Number Divisible by 37: A Comprehensive Guide

Mathematics is a fascinating subject that involves problem-solving, logical reasoning, and number manipulation. One such problem that often comes up is finding the greatest 4-digit number that is exactly divisible by a specific number, in this case, 37. In this article, we will explore the steps to find out the greatest 4-digit number exactly divisible by 37. We will break down the process using various examples and provide a step-by-step guide for understanding the concept.

Understanding the Problem

Letrsquo;s begin by defining the problem clearly. The goal is to find the largest 4-digit number that can be divided by 37 without leaving a remainder. The largest 4-digit number is 9999. Therefore, we need to find the highest multiple of 37 that is less than or equal to 9999. This requires a few simple calculations and some understanding of divisibility rules.

Step-by-Step Approach

Step 1: Identify the Largest 4-Digit Number
First, letrsquo;s identify the largest 4-digit number, which is 9999. This is the number we will start with to find the greatest 4-digit number divisible by 37.

Step 2: Perform the Division
Next, we will perform the division of 9999 by 37 to find out how many times 37 goes into 9999 approximately:

9999 ÷ 37 ≈ 270.0

Step 3: Extract the Integer Part
The result of the division is approximately 270.0, which can be represented in a more precise form as 270.243. The integer part of this division is 270.

Step 4: Multiply to Find the Exact Divisible Number
To find the exact number that is divisible by 37, we take the integer part of the division, which is 270, and multiply it by 37:

270 × 37 9990

Therefore, the greatest 4-digit number that is exactly divisible by 37 is 9990.

Verification

To verify this, letrsquo;s perform the division with 9990 again:

9990 ÷ 37 270 (no remainder)

This confirms that 9990 is indeed divisible by 37 without leaving a remainder.

Alternative Methods

Another method to find the greatest 4-digit number divisible by 37 is to divide 9999 by 37 and round down the result to the nearest whole number:

9999 ÷ 37 ≈ 270.243, rounded down to 270.

Then multiply 270 by 37:

270 × 37 9990

This confirms that 9990 is the greatest 4-digit number divisible by 37.

Conclusion

Using the method described above, we have found that 9990 is the greatest 4-digit number exactly divisible by 37. The steps involved are straightforward and require a good understanding of basic division and divisibility rules. By following these steps, anyone can solve such problems efficiently.

Questions and Answers

Q: Why is 99999 not used in the calculation? A: 99999 is not a 4-digit number. The problem specifically asks for the largest 4-digit number, which is 9999.

Q: What if we want to find the greatest 3-digit number exactly divisible by 37? A: The largest 3-digit number is 999. Dividing 999 by 37 gives approximately 27.027, so the integer part is 27. Multiplying 27 by 37 gives 999, which is exactly divisible by 37.

Q: How can I apply this method to find other numbers divisible by a specific divisor? A: The method is similar. For any number, identify the largest possible number within the specified range, perform the division, extract the integer part, and multiply to find the exact divisible number.

By following these steps and understanding the concept of divisibility, you can solve similar problems involving different divisors and ranges.