Introduction to Divisibility and Finding the Smallest Number

This article delves into the mathematical concept of divisibility and provides a step-by-step guide to solve a common problem: determining the smallest number that must be added to 45 to make it exactly divisible by 12. The methods include traditional division and the use of programming languages like J, making it accessible and informative for a wide audience.

Understanding Divisibility

Divisibility refers to the ability of a number to be divided by another without leaving a remainder. In this context, our goal is to find the smallest integer that, when added to 45, results in a number that is perfectly divisible by 12.

Solution Methodology - Traditional Division

Let's walk through the steps to solve this problem manually:

Step 1: Determine the remainder when 45 is divided by 12.

First, we perform the division:

(45 #247; 12) 3 (whole number part)

Next, we calculate the product of the divisor and the whole number part:

12 #215; 3 36

Now, we find the remainder by subtracting:

45 - 36 9

The remainder is 9, indicating that 45 is 9 more than a multiple of 12.

Step 2: Calculate the amount needed to reach the next multiple of 12.

Since the next multiple of 12 after 36 is 48, we subtract 45 from 48 to find the required amount:

48 - 45 3

Hence, the smallest number that must be added to 45 to make it exactly divisible by 12 is 3.

Using Programming to Solve the Problem

Example using the J programming language:

The output of 3 indicates that the smallest number to be added to 45 for divisibility by 12 is indeed 3.

Brute Force Method

The brute force solution involves identifying the next multiple of 12 after 45. Since 12 #215; 3 36, the next multiple is 12 #215; 4 48. Subtracting 45 from 48 gives:

48 - 45 3

Thus, the smallest number to be added to 45 for exact divisibility by 12 is 3.

Considering Negative Numbers

It's important to note that while -33 is also a valid solution as it makes 12 exactly divisible, the problem usually looks for the smallest positive integer. Additionally, while negative numbers like -45 are technically smaller, the context often restricts the solution to the smallest positive integer.

In summary, the smallest number that must be added to 45 to make it exactly divisible by 12 is 3. This solution can be achieved through traditional division methods or by utilizing programming languages like J.

Conclusion and Further Learning

Mastering such mathematical concepts is not only beneficial for academic purposes but can also enhance problem-solving skills. For those interested in further exploring related topics, resources on divisibility rules, number theory, and mathematical programming languages are highly recommended.