How Can You Quickly Find the Factors of a Number?
Factors are numbers that divide into another number without leaving a remainder. A factor is always less than or equal to the original number.
To find factors quickly:
- Begin with 1 and the number itself.
- Incrementally check each number up to the square root.
- If it divides exactly (no remainder), add both the divisor and the quotient to your list.
If a number has only two factors—1 and itself—it is a prime number. If it has more than two, it is a composite number.
Step-by-Step: Find Factors of 100
Let’s find all the factors of 100.
Start with:
$$ {1, 100} $$Check:
$$ 2 \mid 100 \Rightarrow \text{Add } 2, \frac{100}{2} = 50 $$ $$ {1, 2, 50, 100} $$Check:
$$ 3 \nmid 100 \quad (\text{remainder } = 1) $$Check:
$$ 4 \mid 100 \Rightarrow \text{Add } 4, \frac{100}{4} = 25 $$ $$ {1, 2, 4, 25, 50, 100} $$Check:
$$ 5 \mid 100 \Rightarrow \text{Add } 5, \frac{100}{5} = 20 $$ $$ {1, 2, 4, 5, 20, 25, 50, 100} $$Check:
$$ 6 \nmid 100 \quad (\text{remainder } = 4) $$Check:
$$ 7 \nmid 100 \quad (\text{remainder } = 2) $$Check:
$$ 8 \nmid 100 \quad (\text{remainder } = 4) $$Check:
$$ 9 \nmid 100 \quad (\text{remainder } = 1) $$Check:
$$ 10 \mid 100 \Rightarrow \text{Add } 10 $$ $$ {1, 2, 4, 5, 10, 20, 25, 50, 100} $$
Since $\sqrt{100} = 10$, we stop here.
Final Result
All factors of 100: $$ 1, 2, 4, 5, 10, 20, 25, 50, 100 $$
Highest proper factor:
$$ 50 $$100 is a composite number (more than two factors).
Summary
To find the factors of a number:
- Begin with 1 and the number.
- Check all values up to its square root.
- Add both the divisor and its matching quotient when a division is exact.
The factors of 100 are: $$ 1, 2, 4, 5, 10, 20, 25, 50, 100 $$
100 is a composite number because it has more than two factors.