Forming the Smallest Number from Given Digits: A Comprehensive Guide

When it comes to forming the smallest number from a set of digits, there are specific rules and considerations that must be followed. This guide will help you understand how to form the smallest possible number using the digits 0, 4, 2, and 6, both with and without repetition, and provide a detailed explanation and relevant examples. We will also explore how to handle constraints such as the position of digits and the requirement for the number to be positive.

Understanding the Problem

Given the digits 0, 4, 2, and 6, we need to form the smallest number without repeating any digit. This problem can be broken down into several steps and different scenarios, each with its own set of rules and constraints.

Without Repetition of Digits

The most straightforward approach is to arrange the given digits in ascending order, ensuring that the leading digit is not zero if the number should be a valid four-digit positive number.

Smallest Four-Digit Number

The smallest four-digit number formed from the digits 0, 4, 2, and 6 without repetition is 2046. The reasoning behind this is:

0 cannot be the leading digit as it would make the number invalid (e.g., 0426 is not a four-digit number). Arranging the digits in ascending order gives 2046, which is the smallest valid four-digit number.

Thus, the correct arrangement is 2046

With Repetition of Digits

If repetition of digits is allowed, the problem can take different forms, and we need to consider additional scenarios:

Smallest Four-Digit Number

Without repetition:

The smallest four-digit number is 2046 (as discussed above). However, if repetition is allowed, the smallest number becomes -6420, which is a negative number. But if we are only considering positive whole numbers, the smallest would be 0.246.

Smallest Multi-Digit Numbers

Let's explore the formation of the smallest numbers for various digit lengths:

Three-Digit Numbers

Without repetition: 204 (smallest possible three-digit number without repeating digits). Using the digits 0, 4, and 2: 104 (smallest possible number using all three digits).

Two-Digit Numbers

Without repetition: 24 (smallest possible two-digit number).

One-Digit Numbers

Without repetition: 0 (smallest possible one-digit number). One-digit positive number: 1.

Conclusion

In summary, when forming the smallest number with the digits 0, 4, 2, and 6, the key is to ensure that the number is valid (no leading zeros for four-digit numbers) and is arranged in ascending order. Repetition of digits can significantly change the outcome, leading to different smallest values.