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 efficiently:
- Start with 1 and the number itself.
- Incrementally check numbers up to the square root of the original number.
- If a number divides evenly, include it and its quotient.
If a number has only two factors (1 and itself), it is a prime number. Otherwise, it is a composite number.
Step-by-Step: Find Factors of 26
Let’s go through the process of finding all factors of 26.
Start with:
$$ {1, 26} $$Check:
$$ 2 \mid 26 \Rightarrow \text{Add } 2, \frac{26}{2} = 13 $$
$$ {1, 2, 13, 26} $$Check:
$$ 3 \nmid 26 \quad \text{(remainder = 2)} $$Check:
$$ 4 \nmid 26 \quad \text{(remainder = 2)} $$Check:
$$ 5 \nmid 26 \quad \text{(remainder = 1)} $$Since ( \sqrt{26} \approx 5.1 ), we stop here.
Final Result
All factors of 26:
$$ 1, 2, 13, 26 $$Highest proper factor (not including 26):
$$ 13 $$Since 26 has more than two factors, it is a composite number.
Summary
To find all the factors of a number:
- Start from 1 and test divisibility up to the square root.
- For each divisor, also record the quotient.
The factors of 26 are:
$$ 1, 2, 13, 26 $$
This confirms 26 is a composite number, as it has more than two distinct factors.