# Theory of Whole Numbers: An Introduction with Practice Problems
Whole numbers, also known as non-negative integers, are the basic building blocks of arithmetic and number theory. They are the set of numbers that include zero (0) and all of the positive integers (1, 2, 3, …). Whole numbers are used for counting, ordering, and basic computations that are fundamental to understanding more complex mathematical concepts.
## Understanding Whole Numbers
Whole numbers have several important properties:
– **Closure:** The sum or product of any two whole numbers is also a whole number.
– **Commutative Property:** When adding or multiplying whole numbers, their order can be changed without affecting the result (e.g., `4 + 5 = 5 + 4`).
– **Associative Property:** When adding or multiplying whole numbers, no matter how they are grouped, the result will be the same (e.g., `(2 + 3) + 4 = 2 + (3 + 4)`).
– **Distributive Property:** Multiplication over addition holds true for whole numbers (e.g., `3 * (4 + 5) = (3 * 4) + (3 * 5)`).
– **Identity Elements:** The number 0 is the additive identity because adding 0 to any whole number leaves it unchanged, and the number 1 is the multiplicative identity because multiplying 1 by any whole number leaves it unchanged.
– **Division:** Division of whole numbers may not always result in a whole number. Hence, it is not closed under division.
## Basic Concepts of Whole Numbers
– **Addition:** Combining two or more whole numbers to get another whole number.
– **Subtraction:** Determining how much one whole number is more than another.
– **Multiplication:** Adding a number to itself a certain number of times.
– **Division:** Splitting a number into equal parts.
– **Exponentiation:** Raising a number to the power of another number.
Whole numbers can also be used to explore more advanced topics like prime numbers, composite numbers, factors, multiples, and even basic number theory concepts like divisibility rules and the Fundamental Theorem of Arithmetic.
## Practice Problems
Here are 20 problems exploring the theory of whole numbers, complete with solutions. For brevity, only select solutions will feature written explanations; the rest will present the final answer.
**Problem 1:** What is the sum of the first ten whole numbers?
**Solution 1:** The sum of the first ten whole numbers is obtained by adding them together:
\[
0 + 1 + 2 + … + 9 = 45
\]
**Problem 2:** If you multiply any whole number by 1, what is the result?
**Solution 2:** The result is the number itself, as 1 is the multiplicative identity.
**Problem 3:** Find the product of the first five positive whole numbers.
**Solution 3:** The product is given by:
\[
1 \times 2 \times 3 \times 4 \times 5 = 120
\]
**Problem 4:** How many whole numbers are between 10 and 20?
**Solution 4:** There are 8 whole numbers between 10 and 20 (excluding 10 and 20).
**Problem 5:** What is 2 raised to the 4th power?
**Solution 5:**
\[
2^4 = 16
\]
**Problem 6:** What is the smallest prime whole number?
**Solution 6:** The smallest prime whole number is 2.
**Problem 7:** Calculate the sum: 12 + 23 + 7 + 0.
**Solution 7:** The sum is 42.
**Problem 8:** If you distribute 3 apples to 5 children, how many whole apples does each child get?
**Solution 8:** Each child gets \( \lfloor \frac{3}{5} \rfloor = 0 \) whole apples.
**Problem 9:** What is the next whole number after 999?
**Solution 9:** The next whole number is 1000.
**Problem 10:** How many factors does the number 12 have?
**Solution 10:** The number 12 has 6 factors: 1, 2, 3, 4, 6, 12.
**Problem 11:** Subtract 18 from the smallest 3-digit whole number.
**Solution 11:** The smallest 3-digit whole number is 100, so the result of the subtraction is 82.
**Problem 12:** Identify the largest two-digit prime number.
**Solution 12:** The largest two-digit prime number is 97.
**Problem 13:** What is the fourth multiple of 7?
**Solution 13:** The fourth multiple of 7 is 28.
**Problem 14:** Is 0 a prime number or a composite number?
**Solution 14:** Zero (0) is neither prime nor composite.
**Problem 15:** How many whole number multiples of 5 are there between 1 and 100?
**Solution 15:** There are 19 whole number multiples of 5 between 1 and 100 (excluding 1 and including 100).
**Problem 16:** What is 3 factorial (3!)?
**Solution 16:**
\[
3! = 3 \times 2 \times 1 = 6
\]
**Problem 17:** Calculate the difference between the square of 9 and the square of 5.
**Solution 17:** The difference is
\[
9^2 – 5^2 = 81 – 25 = 56
\]
**Problem 18:** If you list all the whole numbers from 1 to 50, how many times does the digit 3 appear?
**Solution 18:** The digit 3 appears 5 times as a unit (3, 13, 23, 33, 43) and in the tens place once (30-39), giving a total of 15 times.
**Problem 19:** Find the sum of all even whole numbers between 1 and 50.
**Solution 19:** The sum of even numbers between 1 and 50 is 650.
**Problem 20:** Determine whether 49 is a perfect square.
**Solution 20:** Yes, 49 is a perfect square because it is equal to \(7^2\).
Through these problems, it becomes evident that the theory of whole numbers encompasses a wide range of mathematical operations and concepts, forming a foundation for further exploration into mathematics.
