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Whenever we multiply two whole numbers, we get another number. The numbers we multiplied to get the product are called the factors of the product. For example:
- \(2 \times 3 = 6\)
This implies that 2 and 3 are the factors of 6. Another conclusion which we can draw from this is that factors of a number completely divide the number without leaving any remainder.
Example: Let’s consider the number 24. Now, 24 can be divided into factors 6 and 4. Also, 6 can be further factorized into 3 and 2. Moreover, 4 can also be factorized into 2 and 2. So, from this, we can see that the other factors of 24 are 3, 8, and 2. This is because \(12 \times 2 = 8 \times 3 = 3 \times 8 = 24\).
Some Facts about Factors:
- Each and every number has a smallest factor which is 1.
- Every number has a minimum of two factors, which are 1 and the number itself.
- Numbers which have only two factors (1 and the number itself) are called prime numbers.
Prime Factorization
Prime factorization is defined as the product of all the prime factors of a number which, when multiplied together, gives the original number. To represent the prime factors of a number, we may sometimes have to repeat the number.
Example: The factors of 8 are 1 and 2, but to represent 8, we use:
- \(8 = 2 \times 2 \times 2\)
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Some Fun Facts about Factors
- Factors are integers, meaning they are whole numbers and can be positive or negative.
- For any even number, 2 will always be a factor.
- For any number ending in 5, 5 will always be a factor.
- For any number ending in zero, 2, 5, and 10 will always be factors.
- Factoring is commonly used in algebraic expressions to simplify or solve equations.
How to Factor a Number
Follow these steps to factor a number:
- Start by dividing the number by 2.
- If the number is not divisible by 2, try the next prime number (e.g., 3).
- Continue this process until you find a number that divides evenly.
- Represent the factor as a product of all its prime numbers.
Example: To factor 42:
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- So, 42 = 2 × 3 × 7
Exercises for Factoring Numbers
- 12 ⟶
- 19 ⟶
- 34 ⟶
- 41 ⟶
- 14 ⟶
- 35 ⟶
- 24 ⟶
- 63 ⟶
- 50 ⟶
- 70 ⟶
- 12 ⟶ 2 × 2 × 3
Solution:
Step 1: Divide the number by the smallest prime number that is 2 and continue with other prime numbers until the result is not divisible by any prime number.
\(12 ÷ 2 = 6\), \(6 ÷ 3 = 2\)
Step 2: Finally, represent the number as a product of all the prime factors.
\(12 = 2 × 2 × 3\)
- 19 ⟶ 19
Solution:
Step 1: The smallest prime number that 19 is divisible by is 19.
\(19 ÷ 19 = 1\)
Step 2: Therefore, \(19 = 19\).
- 34 ⟶ 2 × 17
Solution:
Step 1: Divide the number by the smallest prime number that is 2 and continue with other prime numbers until the result is not divisible by any prime number.
\(34 ÷ 2 = 17\), \(17 ÷ 17 = 1\)
Step 2: Finally, represent the number as a product of all the prime factors.
\(34 = 2 × 17\)
- 41 ⟶ 41
Solution:
Step 1: The smallest prime number that 41 is divisible by is 41.
\(41 ÷ 41 = 1\)
Step 2: Therefore, \(41 = 41\).
- 14 ⟶ 2 × 7
Solution:
Step 1: Divide the number by the smallest prime number that is 2 and continue with other prime numbers until the result is not divisible by any prime number.
\(14 ÷ 2 = 7\), \(7 ÷ 7 = 1\)
Step 2: Finally, represent the number as a product of all the prime factors.
\(14 = 2 × 7\)
- 35 ⟶ 5 × 7
Solution:
35 is divisible by 5 and 7:
35 = 5 × 7
- 24 ⟶ 2 × 2 × 2 × 3
Solution:
24 is divisible by 2 and 3:
24 = 2 × 2 × 2 × 3
- 63 ⟶ 3 × 3 × 7
Solution:
63 is divisible by 3 and 7:
63 = 3 × 3 × 7
- 50 ⟶ 2 × 5 × 5
Solution:
50 is divisible by 2 and 5:
50 = 2 × 5 × 5
- 70 ⟶ 2 × 5 × 7
Solution:
70 is divisible by 2, 5, and 7:
70 = 2 × 5 × 7