The number 225 is divisible by all its factors, which are 1, 3, 5, 9, 15, 25, 45, 75, and 225.
Understanding what a number is divisible by means identifying all the numbers that can divide it without leaving a remainder. These numbers are also known as its factors. For 225, these factors cover both its prime and composite divisors.
Understanding Divisibility and Factors of 225
When we say a number is "divisible by" another, it means that performing the division results in a whole number with no remainder. For example, 225 divided by 3 equals 75, a whole number, so 225 is divisible by 3.
The Factors of 225
The complete set of positive integers that divide 225 evenly are its factors. These numbers represent all possible whole-number quotients when 225 is divided.
The factors of 225 are:
- 1
- 3
- 5
- 9
- 15
- 25
- 45
- 75
- 225
Here's a table illustrating these factors and their corresponding quotients:
| Divisor (Factor) | Calculation | Quotient |
|---|---|---|
| 1 | 225 ÷ 1 | 225 |
| 3 | 225 ÷ 3 | 75 |
| 5 | 225 ÷ 5 | 45 |
| 9 | 225 ÷ 9 | 25 |
| 15 | 225 ÷ 15 | 15 |
| 25 | 225 ÷ 25 | 9 |
| 45 | 225 ÷ 45 | 5 |
| 75 | 225 ÷ 75 | 3 |
| 225 | 225 ÷ 225 | 1 |
Prime Factors of 225
A special subset of factors are prime factors. These are prime numbers that divide the given number exactly. To find the prime factors, we perform prime factorization:
- Start dividing 225 by the smallest prime number.
- 225 ÷ 3 = 75
- 75 ÷ 3 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
So, the prime factorization of 225 is 3 × 3 × 5 × 5, which can be written as 3² × 5².
The distinct prime factors of 225 are:
- 3
- 5
All other factors of 225 (like 9, 15, 25, 45, 75) are derived from combinations of these prime factors. For instance, 9 is 3×3, and 15 is 3×5.
Practical Insights into Divisibility
Understanding the factors of a number like 225 can be useful in various mathematical contexts, such as simplifying fractions, finding common denominators, or solving algebraic problems.
Recognizing Divisibility Quickly
While finding all factors typically involves some calculation, simple divisibility rules can help identify some factors quickly:
- Divisibility by 3: Sum the digits of the number. If the sum is divisible by 3, the number itself is divisible by 3. For 225, 2 + 2 + 5 = 9. Since 9 is divisible by 3, 225 is divisible by 3.
- Divisibility by 5: If the last digit of the number is 0 or 5, the number is divisible by 5. For 225, the last digit is 5, so it is divisible by 5.
- Divisibility by 9: Similar to the rule for 3, if the sum of the digits is divisible by 9, the number is divisible by 9. For 225, 2 + 2 + 5 = 9. Since 9 is divisible by 9, 225 is divisible by 9.
In summary, 225 is definitively divisible by 1, 3, 5, 9, 15, 25, 45, 75, and 225. These numbers cover all its positive integer factors.
