Finding the Smallest Number Not Divisible by 2, 3, 4, 5, or 6 and Its Digit Sum

Introduction

Given the mathematical problem of identifying the smallest positive integer that is not divisible by 2, 3, 4, 5, or 6, we explore various approaches and the reasoning behind each one. This exploration helps in understanding the intricacies of number theory and divisibility rules, which are fundamental in solving more complex mathematical problems.

Approach 1: Considering Negative Numbers

One might initially consider negative numbers, as any integer multiple of 60 minus 60 will not be divisible by 2, 3, 4, 5, or 6. Examples include -59, -119, -179, and so on. However, it is clear that there is no smallest such number, as subtracting 60 from any negative solution will yield another valid negative solution.

The Case for Non-Negative Numbers

When we restrict ourselves to non-negative numbers, the smallest such number is identified as 1. This is because:

(1) is not divisible by any of the numbers 2, 3, 4, 5, or 6. Therefore, the sum of the digits of 1 is simply 1.

Using Least Common Multiple (LCM) to Find the Smallest Number

The least common multiple (LCM) of 2, 3, 4, 5, and 6 is 60. This means that any number of the form (60k 1) (where (k) is a non-negative integer) is the smallest number that is not divisible by any of these integers. For example:

(60 times 0 1 1) (60 times 1 1 61) (60 times 2 1 121) and so on...

The smallest number in this form is 61, and the sum of its digits is (6 1 7).

Verification of Positivity and Digit Sum

Let us consider (x) to be a positive integer. If (x 60n - 1) (where (n) is a positive integer), then (x) is not divisible by 2, 3, 4, 5, or 6. However, there is no smallest positive integer (n) that satisfies this condition, as subtracting 60 will yield another valid solution. Therefore, the smallest positive integer (x) that is not divisible by any of these numbers is 61, and the sum of its digits is 7.

Conclusion and Further Insight

Are there other number formats that could give the smallest number not divisible by 2, 3, 4, 5, or 6? Indeed, numbers like 7, 11, 101, 1003, etc., are valid, but they are not the smallest. The smallest such number is 7, as it is within a single digit, and the sum of its digits is 7. This conclusion is reached by considering the smallest possible number that is not a multiple of 60 and adjusting it to meet the divisibility criteria.

The digit sum of the smallest number confirms that the problem solution is 7, as both the number 61 and 7 have a digit sum of 7.