How Can You Quickly Find the Factors of a Number?
Factors are the numbers that divide exactly into another number without leaving a remainder. A factor can never be greater than the number itself.
To find factors quickly:
- Start with 1 and the number itself.
- Incrementally check numbers up to the square root of the original number.
- If a number divides evenly, record both it and its quotient.
If a number has only two factors (1 and itself), it’s a prime number. If it has more than two, it’s a composite number.
Step-by-Step: Find Factors of 18
Let’s find all the factors of 18:
Begin with the obvious pair:
$$ {1, 18} $$Check:
$$ 2 \mid 18 \quad \Rightarrow \text{Add } 2, \frac{18}{2} = 9 $$
$$ {1, 2, 9, 18} $$Check:
$$ 3 \mid 18 \quad \Rightarrow \text{Add } 3, \frac{18}{3} = 6 $$
$$ {1, 2, 3, 6, 9, 18} $$Check:
$$ 4 \nmid 18 \quad \text{(remainder = 2)} $$Since ( \sqrt{18} \approx 4.24 ), we can stop here.
Final Result
All factors of 18:
$$ {1, 2, 3, 6, 9, 18} $$Highest proper factor (not including 18):
$$ 9 $$Since 18 has more than two factors, it is a composite number.
Summary
To find all the factors of a number:
- Start from 1 and go up to its square root
- For each divisor that divides evenly, include it and its matching quotient
The factors of 18 are:
$$ 1, 2, 3, 6, 9, 18 $$
This confirms that 18 is a composite number because it has more than two distinct factors.