12.1.1. Factors of natural numbers

You will remember what you learnt about factors in Class VI. Let us take a natural number, say 30, and write it as a product of other natural numbers, say 30 = 2 × 15. We know that 30 can also be written as 3 × 10 = 5 × 6 = 1 × 30. Thus, 1 and 30 are also factors of 30. Thus, 1, 2, 3, 5, 6, 10, 15 and 30 are the factors of 30.

In this chunk, we start by recalling the concept of factors. A factor of a number is any whole number that can be multiplied by another whole number to produce that number. For instance, the number 30 can be expressed as the product of 2 and 15, among other combinations. These products show all the possible pairs of numbers that multiply to give 30. Additionally, 1 and 30 itself are factors of 30, making the complete list of factors for 30 to be 1, 2, 3, 5, 6, 10, 15, and 30. Understanding factors helps in various areas of mathematics, including simplification and factorization of algebraic expressions.

Think of factors as teams of friends working together to achieve a common goal, which in this case is reaching the number 30. Each team represents a pair of factors—like the 2-person team (2 and 15) working together to reach the target of 30. Similarly, a group of three friends could team up (3, 10) or four friends (5, 6) creating other combinations. The number 30 allows many different groups to achieve its value, just as in real life, teams can be formed in multiple ways to accomplish tasks.

Of these, 2, 3 and 5 are the prime factors of 30 (Why?). A number written as a product of prime factors is said to be in the prime factor form; for example, 30 written as 2 × 3 × 5 is in the prime factor form. The prime factor form of 70 is 2 × 5 × 7. The prime factor form of 90 is 2 × 3 × 3 × 5, and so on.

In this chunk, we differentiate between regular factors and prime factors. Prime factors are those that can only be divided by 1 and themselves without leaving a remainder. In our case for number 30, we identify 2, 3, and 5 as prime factors because they meet this criterion. When we express a number as a product of its prime factors, like 30 as 2 x 3 x 5, we take it to its simplest form, known as the prime factor form. Similarly, the prime factor forms for 70 and 90 illustrate this concept further.

Imagine prime factors are like the fundamental building blocks or LEGO pieces that cannot be broken down into smaller pieces. Just as you can build complex structures using basic LEGO bricks, you can create larger numbers using these basic prime factors—the building blocks of mathematics. For example, the number 30 is built by combining these prime LEGO blocks: 2, 3, and 5.

You will notice that 1 is a factor of any number. For example, 1 × 101 = 101. However, when we write a number as a product of factors, we shall not write 1 as a factor, unless it is specially required.

This chunk highlights the unique role of the number 1 in factorization. The number 1 is a universal factor for all natural numbers since any number multiplied by 1 remains unchanged. While it's technically a factor, often for simplicity, we don't include it when listing factors, unless specifically needed. This understanding simplifies our work when factoring larger numbers or algebraic expressions.

Consider the number 1 as a special key that can unlock any door (i.e., any natural number). It’s universally applicable so no matter what number you have, you can always use the key (1) to derive it. Yet, when telling others about the locks you can open (i.e., factors), you don’t typically mention the key. You focus on the unique keys that open different doors.

A number written as a product of prime factors is said to be in the prime factor form; for example, 30 written as 2 × 3 × 5 is in the prime factor form. Similarly, we can express algebraic expressions as products of their factors.

This chunk concludes the section by reaffirming the importance of expressing numbers in their prime factor form and extends the concept to algebraic expressions. Just like numerical factors, algebraic expressions can also be factored into simpler constituent parts. This will be critical when we advance to factorization in algebra, allowing us to break down complex expressions into manageable components.

Think of prime factorization like organizing ingredients for a recipe. Just as you may break down complex recipes into single ingredients (the simplest forms of food), we can break down numbers and algebraic expressions into their 'ingredient' factors. This organization is essential for cooking (solving) and ensures we have a clear path to understanding the final dish (solving an equation or simplifying).

--