Finding the Greatest and Least Four-Digit Numbers Divisible by 6 and 11 Without Using Zero

Divisibility by 6 and 11 is an interesting mathematical challenge. When we are tasked with finding the least and greatest four-digit numbers that satisfy these conditions without using the number zero, the process involves a blend of arithmetic and logical reasoning. This article will guide you through the steps to find these numbers and explain the mathematical principles involved.

Understanding Divisibility Rules

Before we proceed, let's briefly review the basic principles of divisibility:

A number is divisible by 6 if it is divisible by both 2 and 3. A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is a multiple of 11, including 0.

The Least Four-Digit Number

Let's start with the least four-digit number that is divisible by 6 and 11 without using any zeros.

The smallest four-digit number is 1000. However, when we divide 1000 by the least common multiple (LCM) of 6 and 11, which is 66, we find that:

1000/66 15.15... (approximately)

Since we need a whole number, we round it up to the next integer, which is 16. So, we multiply 66 by 16 to get the least four-digit number that is a multiple of 66:

66 * 16 1056

However, 1056 contains a zero, so we need to find the next number without a zero. We continue the series of multiples of 66:

66 * 17 1122

Indeed, 1122 is the smallest four-digit number divisible by 6 and 11, and it does not contain any zeros.

The Greatest Four-Digit Number

Now, let's find the greatest four-digit number that is divisible by 6 and 11 without using any zeros. The largest four-digit number is 9999. Dividing 9999 by the LCM of 6 and 11 (which is 66) gives us:

9999/66 151.5

Again, we need to round up to the next whole number, which is 152.

66 * 152 9936

However, 9936 contains an extra zero, so we need to find the next number in the series without a zero. We can try 66 * 151:

66 * 151 9906

9906 is still not valid as it contains a zero. We can try 66 * 150, but it will be less than four digits. The next option is:

66 * 149 9834

9834 is the greatest four-digit number achievable using the digits 1, 3, 4, and 8 without any zeros and it is divisible by 6 and 11.

Conclusion

In summary, the least four-digit number divisible by 6 and 11 without using any zeros is 1122, and the greatest four-digit number is 9834. This method involves understanding the divisibility rules for 6 and 11, and carefully checking each multiple to ensure it meets the criteria without including any zeros.

Additional Tips

Tips for Future Problems:

Always start from the smallest multiple and adjust to avoid zeros. Use the least common multiple (LCM) of the numbers to find the basic multiples. Check divisibility by 6 and 11 for each candidate number. Ensure no zeros are included in the final answer.

By following these steps, you can tackle similar problems with ease and accuracy.