Factors of 25 | Prime Factorization of 25 - Explained Simply
Today we are going to present here Factor Tree of 25. The factor is the number that divides the original number. The Factors of 25 are 1, 5 and 25 itself.
What are the factors of 25?
The factors of 25 are all the whole numbers that, when divided into 25, result in a whole number with no remainder. We can also think of them as the whole numbers that multiply together to give 25.
Factor Tree Method of 25: Explained Simply
Finding the Factors of 25 - A Step-by-Step Process
To find the factors of 25, we can use a systematic approach, testing numbers one by one to see if they divide 25 evenly. We'll look for "factor pairs" – two numbers that multiply together to give 25.
Start with 1:
Is 1 a factor of 25?
Yes, because 1 * 25 = 25.
This gives us our first factor pair: (1, 25).
Test 2:
Is 2 a factor of 25?
No, because 25 is an odd number, and 2 only divides even numbers evenly.
(25 \ 2 = 12 with a remainder of 1).
Test 3:
Is 3 a factor of 25?
No. (25 \ 3 = 8 with a remainder of 1).
(A quick trick: If the sum of the digits is divisible by 3, the number is divisible by 3. 2+5=7, which is not divisible by 3).
Test 4:
Is 4 a factor of 25?
No. (25 \ 4 = 6 with a remainder of 1).
Test 5:
Is 5 a factor of 25?
Yes, because 5 * 5 = 25.
This gives us our second factor pair: (5, 5).
Important Note: When we find a number that, when multiplied by itself, equals our target number (like 5 × 5 = 25), we've reached the "middle" of our factors. We don't need to test any more numbers up to 25, because any other factor we find will already have its pair identified (e.g., if we had found 2, its pair 12.5 wouldn't be a whole number, but for a number like 12, its factors are 1,2,3,4,6,12. We test up to √12 = 2√3 which is about 3.46. So we test 1, 2, 3. We find 1×12, 2×6, 3×4).
Based on our systematic search, the factors of 25 are:
- 1 (because 1*25 = 25)
- 5 (because 5*5= 25)
- 25 (because 25*1 = 25)
In summary, the factors of 25 are 1, 5 and 25.
Notice that we only list 5 once, even though it appears in the factor pair (5, 5). When listing factors, we list each unique factor only once. Thus, the number 25 has three factors.
All the factors of 25 (1, 5, 25) can be formed by multiplying these prime factors in different combinations (including using no prime factors, which gives 1).
5^0 = 1
5^1 = 5
5^2 = 25
Prime Factorization of 25
Now, let's apply these concepts to our target number: 25.
First, let's determine if 25 is prime or composite.
Can 25 be divided evenly by any number other than 1 and 25?
Yes, it can be divided by 5.
Therefore, 25 is a composite number. This means we can find its prime factors!
We'll use two common methods to find the prime factorization of 25.
The Factor Tree Method
The factor tree method involves breaking down the number into any two factors, then continuing to break down those factors until all "branches" end in prime numbers.
Start with the number 25 at the top:
```
25
/ \
```
Find any two factors of 25 (other than 1 and 25):
The most obvious pair is 5 and 5.
```
25
/ \
5 5
```
Check if these factors are prime:
Is 5 a prime number?
Yes, its only factors are 1 and 5.
Since both branches have ended in prime numbers, we stop.
Collect the prime factors:
The prime factors at the ends of the branches are 5 and 5.
So, the prime factorization of 25 is 5 × 5.
The Division Method (Repeated Division)
The division method involves repeatedly dividing the number by the smallest possible prime number until the result is 1.
Start with the number 25.
Find the smallest prime number that divides 25 evenly:
Can 25 be divided by 2?
No (it's not an even number).
Can 25 be divided by 3?
No (2 + 5 = 7, which is not divisible by 3).
Can 25 be divided by 5? Yes!
```
5 | 25
|5
```
Now take the result (5) and find the smallest prime number that divides it evenly:
Can 5 be divided by 2? No.
Can 5 be divided by 3? No.
Can 5 be divided by 5? Yes!
```
5 | 25
5 | 5
|1
```
Stop when the result is 1.
Collect all the prime divisors you used:
The prime divisors on the left are 5 and 5.
So, the prime factorization of 25 is 5 × 5.
Both methods lead us to the same unique prime factorization:
25 = 5 × 5
This can also be written using exponents for a more compact form:
25 = 5² (read as "five squared" or "five to the power of two")
What does this mean? It means that the number 25 is fundamentally built from two prime number 5s multiplied together. There's no other combination of prime numbers that will multiply to give you 25.