Problem 1 - A Gentle Introduction to the Art of Mathematics

Question

An integer \(n\) is doubly-even if it is even, and the integer \(m\) guaranteed to exist because \(n\) is even is itself even. Is 0 doubly-even? What are the first 3 positive, doubly-even integers?

Step-by-Step Solution

Answer

0 is doubly-even. The first 3 positive, doubly-even integers are 4, 8, and 12.

Step 1: Understanding Doubly-Even

An integer is doubly-even if it is even, and the quotient obtained when it is divided by 2 is also an even number.

Step 2: Check if 0 is Doubly-Even

First, check if 0 is even. Since 0 divided by 2 equals 0, and 0 is an integer, it is even. Then, check if the quotient (0) is even. Since 0 is even, 0 is doubly-even.

Step 3: Finding the First Positive Doubly-Even Integer

Start with the smallest positive even number, 2. Dividing 2 by 2 gives 1, which is not even, so 2 is not doubly-even. Next is 4. Dividing 4 by 2 gives 2, which is even. Therefore, 4 is the first positive, doubly-even integer.

Step 4: Identifying the Next Positive Doubly-Even Integers

Continue this process: 6 divided by 2 equals 3, which is not even. 8 divided by 2 equals 4, which is even. Hence, 8 is doubly-even. Similarly, for 12: 12 divided by 2 equals 6, which is even. Therefore, 12 is also doubly-even.

Key Concepts

Understanding Even Numbers

Even numbers are integers that can be exactly divided by 2 without leaving a remainder. For example, numbers like 0, 2, 4, and 6 are all even because when they are divided by 2, the result is an integer (0, 1, 2, and 3 respectively).
Watches out for numbers like 1, 3, and 5, which cannot be divided by 2 without leaving a fraction; these are called odd numbers.
Even numbers have a distinctive property that makes them useful in various mathematical contexts, including our problem on doubly-even integers.

Exploring Integer Division

Integer division is a concept where one integer is divided by another, and the result is a whole number, without any fractional part. For instance, when we divide 8 by 2, the result is 4, an integer.
However, if we divide 7 by 2, the result would be 3.5, which is not an integer. In integer division, we disregard the decimal part and only consider the whole number.
This is especially important in our doubly-even problem because it requires us to focus strictly on the whole numbers after division.

Mathematical Problem-Solving

Solving mathematical problems often involves breaking down the problem into understandable parts. Let's apply this to our doubly-even integers.
First, comprehend the term doubly-even: an integer is doubly-even if it is even, and its division by 2 is also even. For instance, to see if 8 is doubly-even, we start by checking if it's even (Yes, 8 divided by 2 equals 4).
Then, we check if 4 (the quotient) is also even (Yes, 4 divided by 2 equals 2, another even number).
By breaking the problem into steps—checking each part independently—we can accurately determine whether a number meets all the criteria to be doubly-even.