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Substitution is a powerful technique in algebra and number theory. To understand substitution, think of replacing a variable with another value or expression to simplify a problem. In our example, we used substitution to prove divisibility. Here's how:

First, we identified that if 12 divides n, then n can be written as a multiple of 12. So, we wrote n as 12k where k is an integer. This substitution helped simplify the problem.

Next, we knew that n divides 4m. By substituting n with 12k, we transformed the equation 4m = nq into 4m = 12kq. This substitution made the equation easier to work with, leading us to the next steps in proving divisibility.

Substitution helps break down complex problems into simpler parts by replacing variables with known values or expressions.

Divisibility rules are guidelines that help us determine if one number is divisible by another without performing long division. In this problem, we focus on the rules for 12 and 3.

For 12 to divide a number, that number must be divisible by both 3 and 4. This is because 12 is the product of these two numbers. So, any number divisible by 12 is automatically divisible by 3 and 4.

In the problem, we found that 12 divides n, indicating that n is a multiple of 12. When simplifying the equation to m = 3kq, we observed that m is written as a product of 3, which means m is divisible by 3.

Understanding these rules simplifies checking for divisibility quickly, aiding in solving complex number problems in an efficient manner.

Integer properties are essential in various areas of mathematics, especially in number theory and algebra. These properties include understanding integers' behavior under different operations such as addition, multiplication, and division.

In our exercise, we worked with the properties of integers that relate to divisibility. We started with two integers, n and m, and leveraged the fact that if an integer divides another, their relationship can be expressed through multiplication with another integer (e.g., n = 12k, where k is an integer).

Later, the equation 4m = nq was simplified using integer properties. By substituting n = 12k, we got 4m = 12kq. Dividing both sides by 4, we found m = 3kq. This showed that m is an integer multiple of 3, hence proving that 3 divides m.

Being familiar with these properties allows us to manipulate and simplify equations effectively, leading to clear and logical proofs.