1.1 Introduction to Whole Numbers

Whole numbers form the foundation of our number system. They're the building blocks we use for counting, measuring, and everyday math. Understanding place value, multiples, and divisibility rules helps you work with these numbers more efficiently.

Prime factorization breaks numbers down to their basic components. This skill is crucial for finding common factors and multiples, which come in handy for simplifying fractions and solving real-world problems involving groups and schedules.

Understanding Whole Numbers

Place value in whole numbers

Every digit in a whole number has a value determined by its position. Positions increase by powers of 10 as you move from right to left: ones, tens, hundreds, thousands, and so on.

Whole numbers can be written in three forms:

• Standard form: 1,234
• Word form: one thousand two hundred thirty-four
• Expanded form: $1000 + 200 + 30 + 4$

To compare or order whole numbers, start at the leftmost digit and work right. The number with a greater value in the leftmost place is larger. If those digits are equal, move to the next place value and compare again. For example, 4,712 vs. 4,698: the thousands digit is the same (4), but the hundreds digit is 7 vs. 6, so 4,712 is larger.

A number line is a helpful way to visualize whole numbers and see how they relate to each other. In everyday life, you'll see whole numbers used for money, measurements (length, weight, time), population counts, and inventory quantities.

Multiples and divisibility rules

Multiples are the products you get when you multiply a number by 1, 2, 3, and so on. For example, the first five multiples of 3 are 3, 6, 9, 12, and 15.

Divisibility rules give you a quick way to check whether one number divides evenly into another, without doing long division:

• By 2: Last digit is even (0, 2, 4, 6, or 8)
• By 3: The sum of all digits is divisible by 3 (e.g., 141 → $1 + 4 + 1 = 6$, and 6 is divisible by 3, so 141 is too)
• By 4: The last two digits form a number divisible by 4 (e.g., 316 → 16 is divisible by 4)
• By 5: Last digit is 0 or 5
• By 6: Divisible by both 2 and 3
• By 9: The sum of all digits is divisible by 9
• By 10: Last digit is 0

Types of whole numbers

• Natural numbers (also called counting numbers) start from 1: 1, 2, 3, 4, ...
• Whole numbers include all natural numbers plus zero: 0, 1, 2, 3, 4, ...
• Even numbers are divisible by 2 (0, 2, 4, 6, ...)
• Odd numbers are not divisible by 2 (1, 3, 5, 7, ...)

The distinction between natural numbers and whole numbers is small but worth remembering: whole numbers include zero, natural numbers don't.

Prime Factorization and Least Common Multiples

Prime factors and common multiples

A prime number has exactly two factors: 1 and itself. The first several primes are 2, 3, 5, 7, 11, 13, and 17. Notice that 2 is the only even prime number.

A composite number has more than two factors. Examples: 4, 6, 8, 9, 10, 12. The number 1 is neither prime nor composite.

Prime factorization breaks a composite number into a product of primes. Here's how to do it:

1. Divide the number by the smallest prime that goes into it evenly.
2. Take the result and divide again by the smallest prime that works.
3. Keep going until the quotient itself is prime.
4. Write the result using exponents for any repeated factors.

For example, with 36:

• $36 \div 2 = 18$
• $18 \div 2 = 9$
• $9 \div 3 = 3$ (3 is prime, so you're done)
• Result: $36 = 2^2 \times 3^2$

The greatest common factor (GCF) is the largest factor shared by two or more numbers. To find it, write the prime factorization of each number, then multiply the common prime factors using the lowest exponent of each.

The least common multiple (LCM) is the smallest positive number that is a multiple of two or more numbers. To find it, write the prime factorization of each number, then multiply all prime factors using the highest exponent of each.

You can also use this formula: $LCM(a, b) = \frac{a \times b}{GCF(a, b)}$

For example, find the LCM of 12 and 18:

• $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$
• Take the highest power of each prime: $2^2 \times 3^2 = 36$
• So $LCM(12, 18) = 36$

Rounding and estimation are also useful when you need approximate values rather than exact answers, such as quickly checking whether your calculation is reasonable.