Question

Prove that every number greater than 7 is a sum of a nonnegative integer multiple of 3 and a nonnegative integer multiple of 5 .

Short answer

Every integer greater than 7 can be written as 3x+5y for nonnegative x and y.

Step-by-step solution

Understanding the Problem

We need to show that any integer greater than 7 can be expressed as a combination of multiples of 3 and 5, where these multiples are nonnegative integers.

Forming an Approach

To solve this problem, we'll demonstrate the statement through mathematical induction. We'll first verify that there are base cases greater than 7 that satisfy the condition. Then, we'll show that if the condition holds for some integer, it must hold for the next integer.

Base Case Verification

Check integers greater than 7, such as 8, 9, 10, 11, and 12, and verify they can be written as combinations of multiples of 3 and 5: - 8 = 3×1 + 5×1 - 9 = 3×3 + 5×0 - 10 = 3×0 + 5×2 - 11 = 3×2 + 5×1 - 12 = 3×4 + 5×0 All these numbers can be expressed as described.

Inductive Step Setup

Assume for some integer k (where k > 7), the number k can be expressed as 3a + 5b, where a and b are nonnegative integers. We will then show it holds for k + 1.

Prove Inductive Step

To prove for k+1: - If k = 3a + 5b and a > 0, then k + 1 = 3(a - 1) + 5(b + 1) - If a = 0 in any case, note that as long as k is greater than 7, we already proved base cases up to 12 and these recur every few steps. Thus if k = 8 through 12, k+1 is similarly expressible. Hence, any k+1 is expressible as desired.

Key concepts

Integer Representation

Understanding integer representation is crucial when dealing with mathematical proof, especially in mathematical induction. When we talk about integer representation, we refer to expressing a particular integer as a combination of other integers multiplied by coefficients. In our exercise, any integer greater than 7 can be expressed as a sum of nonnegative integer multiples of 3 and 5.

For instance, when you encounter a number like 11, you can represent it as \(3 \times 2 + 5 \times 1\). Here, 3 and 5 are the base numbers, while 2 and 1 are the nonnegative integer coefficients. This approach helps in visualizing how a given number fits into a pattern described by the rule or formula we are working with.

This form of representation is essential because it sets the foundation for establishing general proof for all numbers in a sequence. It also allows mathematicians to verify base cases or specific instances, and then infer about a broader set of numbers.

Nonnegative Integers

Nonnegative integers are the integers that are either zero or positive, such as 0, 1, 2, 3, and so on. In the context of our exercise, these are the coefficients we use when expressing any integer greater than 7 as a combination of multiples of 3 and 5.

Why nonnegative? This becomes relevant when formulating problems in mathematical induction. Using nonnegative integers ensures that we only incorporate viable multiples, thus avoiding complexities like negative results or undefined conditions. These integers generate more straightforward and logically coherent representations that align with the problem requirements.

• Nonnegative values provide a real-world applicable measure, as negative multiples do not make sense when discussing countable objects like money or steps.
• Using these values ensures every integer is expressible by having enough positive additions available through multiples.

By constraining factors to nonnegative integers, it is easier to maintain consistency in the problem-solving process, ensuring reliability in mathematical proofs.

Base Case Verification

In mathematical induction, verifying the base case is one of the crucial initial steps. The importance of base case verification lies in its role as the foundational truth upon which the entire inductive proof builds.

When proving that every number greater than 7 is a sum of nonnegative integer multiples of 3 and 5, we began by validating specific instances known as base cases. For example, numbers like 8, 9, 10, 11, and 12 were checked:

• 8 can be expressed as \(3 \times 1 + 5 \times 1\)
• 9 is \(3 \times 3 + 5 \times 0\)
• 10 as \(3 \times 0 + 5 \times 2\)
• 11 is \(3 \times 2 + 5 \times 1\)
• 12 as \(3 \times 4 + 5 \times 0\)

Confirming these examples proves that the initial statement holds true for these specific cases. Once affirmed, these serve to validate the logic applied in further steps of mathematical induction.

Verifying base cases consolidates the assumption that the proposition is initially accurate, serving as the stepping stone for proving its universality. It ensures that no errors are present before asserting the statement's continuity across all successive integers in the sequence.