Lesson 1.3: Whole Numbers

New Concept

In this lesson, we'll explore whole numbers as combinations of their prime factors. This 'building block' approach is key to finding the Greatest Common Factor (GCF) and Lowest Common Multiple (LCM) for solving various mathematical problems.

What’s next

Next, we will break down these ideas with interactive examples on prime factorization, GCF, and LCM. You'll then apply these skills in practice sets.

Property

A counting number (larger than 1) that is evenly divisible only by 1 and itself is called a prime number. A counting number larger than 1 that is not prime is called composite.

Examples

• The number 17 is prime because its only factors are 1 and 17.
• The number 25 is composite because it is divisible by 5, in addition to 1 and 25. Its factors are 1, 5, and 25.
• The number 2 is the smallest prime number and the only even prime. All other even numbers are composite because they are divisible by 2.

Explanation

Think of prime numbers as basic building blocks; they only have two factors, 1 and themselves. Composite numbers are built from primes and can be divided by more than just 1 and themselves. The number 1 is neither prime nor composite.

Property

The list of prime numbers whose product is the original number is called the prime factorization of a composite number. To find the prime factorization of a larger number, we can use a factor tree. Every whole number greater than 1 can be written in exactly one way as a prime number or as a product of primes (Fundamental Theorem of Arithmetic).

Examples

• To find the prime factorization of 48, we can use a factor tree. Start with $48 = 6 \times 8$. Then $6 = 2 \times 3$ and $8 = 2 \times 2 \times 2$. The prime factorization is $48 = 2 \times 2 \times 2 \times 2 \times 3$.
• Find the prime factorization of 90. We can start with $90 = 9 \times 10$. Breaking these down gives $9 = 3 \times 3$ and $10 = 2 \times 5$. So, the prime factorization of 90 is $2 \times 3 \times 3 \times 5$.
• Let's find the prime factorization for 56. We can write $56 = 7 \times 8$. Since 7 is prime, we only factor 8 as $8 = 2 \times 2 \times 2$. The complete prime factorization is $56 = 2 \times 2 \times 2 \times 7$.

Explanation

Prime factorization is like finding a number's unique secret code. Every composite number has one specific set of prime numbers that, when multiplied, equal that number. A factor tree is a visual way to break a number down into its prime factors.

Property

The greatest common factor or GCF of two whole numbers is the largest factor of both numbers. To find the GCF using prime factorization, multiply together all the prime factors that appear in the factorization of both numbers.

Examples

• To find the GCF of 20 and 30, list their factors. Factors of 20 are {1, 2, 4, 5, 10, 20} and factors of 30 are {1, 2, 3, 5, 6, 10, 15, 30}. The largest factor in both lists is 10.
• Find the GCF of 42 and 56 using prime factorization. We have $42 = 2 \times 3 \times 7$ and $56 = 2 \times 2 \times 2 \times 7$. The common factors are one 2 and one 7. So, the $\text{GCF} = 2 \times 7 = 14$.
• Let's find the GCF of 18 and 45. The prime factorizations are $18 = 2 \times 3 \times 3$ and $45 = 3 \times 3 \times 5$. The common prime factors are two 3s. Therefore, the $\text{GCF} = 3 \times 3 = 9$.

Explanation

The GCF is the largest number that divides into two or more numbers without leaving a remainder. It's like finding the biggest identical group you can make from different sets of items. This is very useful for simplifying fractions.

Property

The lowest common multiple or LCM of two whole numbers is the smallest number that is a multiple of both whole numbers. To find the LCM using prime factorization, use each factor the most times it appears in either one of the two given numbers and multiply them together.

Examples

• To find the LCM of 9 and 12, list their multiples. Multiples of 9 are 9, 18, 27, 36, ... Multiples of 12 are 12, 24, 36, ... The smallest number on both lists is 36.
• Find the LCM of 10 and 15 using prime factorization. The factorizations are $10 = 2 \times 5$ and $15 = 3 \times 5$. We need one 2, one 3, and one 5. The $\text{LCM} = 2 \times 3 \times 5 = 30$.
• Let's find the LCM of 8 and 14. The prime factorizations are $8 = 2 \times 2 \times 2$ and $14 = 2 \times 7$. We need three 2s (from 8) and one 7 (from 14). The $\text{LCM} = 2 \times 2 \times 2 \times 7 = 56$.

Explanation

The LCM is the smallest positive number that is a multiple of two or more numbers. It's useful for finding when two things with different cycles will happen at the same time, like buses arriving or gears aligning.