Question

Prove that \(\\{12 a+25 b: a, b \in \mathbb{Z}\\}=\mathbb{Z}\).

Short answer

The equation \(12a + 25b = n\) holds true for any integer \(n\) which proves that \(\{12 a+25 b: a, b \in \mathbb{Z}\}=\mathbb{Z}\).

Step-by-step solution

Find the Case When K Equals 1

First, consider the equation \(12a + 25b = 1\). It can be observed that if \(a = 2\) and \(b = -1\), the equation holds i.e. \(12*2 + 25*(-1) = 1\). Hence, there exists at least one element K=1 which can be written as \(12a + 25b\).

Show that the sum of any integer with K is also in the same form

Consider an integer \(n\) and let the equation be \(1 + n = 12a + 25b\). The integer \(n\) can be expressed as \(12*x\) for any \(x \in \mathbb{Z}\). Hence, adding n to 1 gives us: \(1 + 12x = 12*a + 25*b\), \(12(2+x) + 25*-1 = 12*a + 25*b\). Now it can be seen when \(m = 2+x\) and \(n = -1\), the resulting equation holds. Hence, for all \(n \in \mathbb{Z}\), they exist integers m and n such that \(K + n = 12m + 25n\).

Conclusion

Since for K=1 a explicit representation as \(12a + 25b\) was found and for any \(n \in \mathbb{Z}\) the sum with K can also be represented in the form \(12a + 25b\), it can be concluded that \(\{12 a+25 b: a, b \in \mathbb{Z}\}=\mathbb{Z}\)

Key concepts

Number Theory

Number theory is a branch of pure mathematics that deals with the properties and relationships of integers. It invites students to explore the intricate patterns and structures within the set of whole numbers. Understanding the concepts within number theory often involves delving into prime numbers, divisibility rules, and the greatest common divisors among other topics.

In the context of the given exercise, number theory is applied to demonstrate a relationship between integer combinations. Specifically, the exercise proves that any integer can be represented as a linear combination of the numbers 12 and 25. Here, the concept of linear combinations is integral; it means that any integer can be expressed as some multiple of 12 added to some multiple of 25. The proof relies on the ability to represent the smallest positive integer in this form and then extend the argument to all integers using number theory principles.

Integer Combinations

The theorem dealing with integer combinations often falls under the umbrella of number theory. In essence, an integer combination implies that integers can be combined using addition and multiplication (by other integers) to produce another integer. This concept is beautifully illustrated in our exercise, where we show that every integer can be expressed as a combination of multiples of 12 and 25.

The exercise's Step 1 starts by finding particular values of 'a' and 'b' such that 12 times 'a' plus 25 times 'b' equals 1, the smallest positive integer. From this starting point, Step 2 demonstrates how this principle can be used to express any integer 'n' in the same form by adding 'n', which can itself be written as a multiple of 12. Thus, integer combinations allow us to build up to any number from a base case.

Algebraic Proofs

Algebraic proofs are logical arguments that utilize algebraic techniques to demonstrate the truth of a given statement. They require a clear understanding of algebraic expressions, equations, and properties of equality and inequality.

For the exercise at hand, an algebraic proof is used to validate that the set of all combinations of the form 12a + 25b indeed covers every integer. Step 1 of the solution provides a specific instance, a base case, showing that 1, an integer, can be written in this form. Step 2 generalizes this result, proving that if 1 can be expressed as such, then by adding a multiple of 12 (12x), any integer 'n' can be achieved in the same form. This is a classic algebraic proof technique known as 'generalization from the base case'. Through clear logical steps and algebraic manipulation, the proof establishes the desired set equivalence.