The prime factorization of 51 is 3×173 cross 17 3×17 .
An Extensive Guide to Prime Factorization
What is Prime Factorization?
Prime factorization is the process of breaking down a composite number into its constituent prime numbers. When these prime numbers are multiplied together, they equal the original number. The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number itself or can be represented as the product of a unique set of prime numbers, regardless of the order of the factors. This uniqueness is what makes prime factorization a fundamental concept in number theory.
Understanding Prime and Composite Numbers
- Prime Number: A natural number greater than 1 that has only two distinct positive divisors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and so on.
- Composite Number: A natural number greater than 1 that is not prime. This means it has more than two positive divisors. For example, 6 is a composite number because its divisors are 1, 2, 3, and 6.
The Process of Prime Factorization: Step-by-Step
The most common method for finding the prime factorization of a number is through repeated division by prime numbers.
### Step 1: Divide the number by the smallest possible prime number.
Start with the smallest prime number, which is 2. Check if the number is divisible by 2. If it is, perform the division. If not, move to the next smallest prime number, which is 3.
Let's apply this to the number 51.
-
Is 51 divisible by 2? No, because it is an odd number.
-
Is 51 divisible by 3? To check for divisibility by 3, you can sum the digits of the number. If the sum is divisible by 3, the number itself is divisible by 3. The digits of 51 are 5 and 1, and their sum is 5+1=65 plus 1 equals 6
5+1=6
. Since 6 is divisible by 3, we know that 51 is also divisible by 3.
51÷3=1751 divided by 3 equals 17
51÷3=17
The result is 17.
### Step 2: Repeat the process with the new quotient.
Now we take the quotient from the previous step, which is 17, and repeat the process of trying to divide it by prime numbers.
-
Is 17 divisible by 2? No.
-
Is 17 divisible by 3? No, 1+7=81 plus 7 equals 8
1+7=8
, which is not divisible by 3.
-
Is 17 divisible by 5? No, it doesn't end in 0 or 5.
-
Is 17 divisible by 7? No, 7×2=147 cross 2 equals 14
7×2=14
and 7×3=217 cross 3 equals 21
7×3=21
.
-
We continue testing primes until we find one that divides 17. However, at this point, we can recognize that 17 is a prime number itself, meaning its only positive divisors are 1 and 17.
Since 17 is a prime number, the process stops here. The prime factors of 51 are the prime numbers we divided by and the final prime number we were left with.
### Step 3: Write the prime factors using exponents.
The prime factors we found are 3 and 17. When we write the prime factorization, we multiply these factors together:
3×173 cross 17
3×17
If a prime factor is repeated, such as in the factorization of 12, which is 2×2×32 cross 2 cross 3
2×2×3
, we use exponents to write it more concisely. The repeated factor, 2, would be written as 222 squared
22
. Therefore, the prime factorization of 12 is 22×32 squared cross 3
22×3
.
In the case of 51, the prime factors 3 and 17 each appear only once, so their exponents are both 1. The factorization is written as **3×173 cross 17
3×17** .
Answer:
The prime factorization of 51 is **3×173 cross 17
3×17** .
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