How Can You Quickly Find the Factors of a Number?
Factors are numbers that divide evenly into another number—without leaving a remainder. No factor can be greater than the number itself.
To find factors efficiently:
- Start with 1 and the number itself.
- Incrementally test integers up to the square root of the number.
- If a number divides evenly, include both it and its quotient.
A number with exactly two factors—1 and itself—is a prime number. A number with more than two factors is a composite number.
Step-by-Step: Find Factors of 81
We’ll now find all factors of 81 using the method described above:
Begin with:
$$ {1, 81} $$Test:
$$ 2 \nmid 81 \quad \text{(remainder = 1)} $$Test:
$$ 3 \mid 81 \Rightarrow \text{Add } 3, \frac{81}{3} = 27 $$
$$ {1, 3, 27, 81} $$Test:
$$ 4 \nmid 81 \quad \text{(remainder = 1)} $$Test:
$$ 5 \nmid 81 \quad \text{(remainder = 1)} $$Test:
$$ 6 \nmid 81 \quad \text{(remainder = 3)} $$Test:
$$ 7 \nmid 81 \quad \text{(remainder = 4)} $$Test:
$$ 8 \nmid 81 \quad \text{(remainder = 1)} $$Test:
$$ 9 \mid 81 \Rightarrow \text{Add } 9 $$
$$ {1, 3, 9, 27, 81} $$
Since ( \sqrt{81} = 9 ), we stop here.
Final Result
All factors of 81:
$$ 1, 3, 9, 27, 81 $$Highest proper factor:
$$ 27 $$Since 81 has more than two factors, it is a composite number.
Summary
To find the factors of a number:
- Start from 1 and test up to the square root of the number.
- For each divisor, include both the divisor and its quotient.
The factors of 81 are:
$$ 1, 3, 9, 27, 81 $$
81 is a composite number because it has more than two factors.