Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic (also known as the unique factorization theorem) is a cornerstone of number theory. It states that every integer greater than 1 can be uniquely represented as a product of prime numbers, up to the order of the factors. This means that, no matter how you try to break down a number into its prime factors, you’ll always get the same prime numbers in the end (though possibly in a different order).

Formulae

The general form of the prime factorization of a number *n* can be expressed as:

$$n = p_1^{a_1} \times p_2^{a_2} \times … \times p_k^{a_k}$$

Where:

• *n* is the integer greater than 1.
• *p₁, p₂, …, p_k* are distinct prime numbers.
• *a₁, a₂, …, a_k* are positive integers representing the exponents of the prime factors.

Examples

Here are a couple of examples illustrating prime factorization:

Example-1: Factorize the number 12.

12 can be divided by 2: 12 / 2 = 6

6 can be divided by 2: 6 / 2 = 3

3 can be divided by 3: 3 / 3 = 1

Therefore, the prime factorization of 12 is: $$2^2 \times 3$$

Example-2: Factorize the number 35.

35 can be divided by 5: 35 / 5 = 7

7 can be divided by 7: 7 / 7 = 1

Therefore, the prime factorization of 35 is: $$5 \times 7$$

Theorem with Proof

Theorem: Every integer greater than 1 can be written as a product of prime numbers, and this factorization is unique up to the order of the factors.

Proof:

Existence (Showing that factorization exists): We’ll use the method of strong induction.

1. Base Case: The number 2 is prime. So, it can be represented as a product of a single prime factor (itself).
2. Inductive Step: Assume that every integer greater than 1 and less than *n* (where *n* > 2) can be written as a product of primes. We now want to show that *n* can also be written as a product of primes.
• If *n* is prime, then it is a product of primes (itself).
• If *n* is composite (not prime), then there exist integers *a* and *b* such that *n* = *a* * b*, where 1 < *a* < *n* and 1 < *b* < *n*. By the inductive hypothesis, both *a* and *b* can be written as products of primes. Therefore, *n* = *a* * b* can also be written as a product of primes.
3. Conclusion: By induction, every integer greater than 1 can be written as a product of primes.

Uniqueness (Showing that factorization is unique): We’ll use proof by contradiction.

1. Assume that there exists an integer *n* > 1 that has two distinct prime factorizations: $$n = p_1 \times p_2 \times … \times p_r = q_1 \times q_2 \times … \times q_s$$ where *p_i* and *q_j* are prime numbers.
2. Without loss of generality, assume *p₁* is different from all *q_j*.
3. Since *p₁* divides the left side of the equation, it must also divide the right side. However, a prime number only divides a product of numbers if it divides at least one of the numbers in the product. Thus, *p₁* must divide at least one of the *q_j*.
4. But since all *q_j* are prime, and *p₁* is different from all *q_j*, *p₁* cannot divide any *q_j*. This is a contradiction.
5. Therefore, our initial assumption that *n* has two distinct prime factorizations must be false. Hence, the prime factorization is unique up to the order of the factors.

Common mistakes by students

• Forgetting to continue factoring: Students sometimes stop factoring a number before reaching all prime factors. For instance, they might stop at `4 x 3` for the number 12, forgetting to further break down 4.
• Including 1 as a prime factor: Students incorrectly list 1 as a prime number. 1 is not considered a prime number because it only has one factor (itself).
• Misidentifying prime numbers: Students may mistakenly identify composite numbers (numbers that are not prime) as prime numbers.

Real Life Application

The Fundamental Theorem of Arithmetic has practical applications in various fields:

• Cryptography: Prime factorization is fundamental to modern cryptography. The difficulty of factoring large numbers into primes is the basis for the security of many encryption algorithms, like RSA.
• Computer Science: Used in hashing algorithms, generating unique identifiers, and optimization problems.
• Scheduling and Planning: Used in calculating the least common multiple (LCM) to find the optimal time for events that occur at regular intervals.
• Construction and Design: Architects and engineers may use prime factorization in order to determine the optimal size or ratios in the designs.

Fun Fact

The search for large prime numbers is an ongoing area of research. The largest known prime number (as of this writing) has millions of digits! The Great Internet Mersenne Prime Search (GIMPS) is a distributed computing project that focuses on finding these massive prime numbers.

Recommended YouTube Videos for Deeper Understanding

Q.1 What is the prime factorization of 120?

Check Solution

Ans: B

$120 = 2 \times 60 = 2 \times 2 \times 30 = 2 \times 2 \times 2 \times 15 = 2 \times 2 \times 2 \times 3 \times 5 = 2^3 \times 3 \times 5$

Q.3 If the prime factorization of a number is $2^4 \times 3^2 \times 5$, what is the number?

Check Solution

Ans: C

$2^4 \times 3^2 \times 5 = 16 \times 9 \times 5 = 144 \times 5 = 720$

Q.4 The prime factorization of a number $n$ is $2^a \times 3^b \times 7^c$. If $n = 252$, what are the values of $a$, $b$, and $c$?

Check Solution

Ans: B

$252 = 2 \times 126 = 2 \times 2 \times 63 = 2 \times 2 \times 3 \times 21 = 2 \times 2 \times 3 \times 3 \times 7 = 2^2 \times 3^2 \times 7$. Therefore a=2, b=2, c=1

Q.5 The number 360 can be written as a product of primes $2^x \times 3^y \times 5^z$. What is the value of $x + y + z$?

Check Solution

Ans: D

$360 = 2 \times 180 = 2 \times 2 \times 90 = 2 \times 2 \times 2 \times 45 = 2^3 \times 3 \times 15 = 2^3 \times 3^2 \times 5$. So $x=3$, $y=2$, and $z=1$. Thus, $x + y + z = 3+2+1 = 6$.

Next Topic: HCF and LCM: Relationship & Applications