The fundamental theorem of arithmetic — lesson. Mathematics State Board, Class 10.

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Theorem:

Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Explanation:

Let us take the composite number

$N$ .

Decompose the number

$N$ into the product of primes.

Here, the number

$N = x_1 \times x_2$ . But, both

$x_1$ and

$x_2$ are again composite numbers. So, factorise it further to obtain a prime number.

The prime factors of

$x_1 = p_1 \times p_2$ .

The prime factors of

$x_2 = p_3 \times p_4$ .

We get,

$N = p_1 \times p_2 \times p_3 \times p_4$ where

$p_1$ ,

$p_2$ ,

$p_3$ and

$p_4$ are all prime numbers.

If we have repeated primes in a product, then we can write it as a power.

In general, given a composite number

$N$ , we factorise it uniquely in the form

$N = {p_{1}}^{q_{1}} \times {p_{2}}^{q_{2}} \times {p_{3}}^{q_{3}} \times ... \times {p_{n}}^{q_{n}}$ where

$p_{1}, p_{2}, p_{3}, ... p_{n}$ are prime numbers, and

$q_{1}, q_{2}, q_{3}, ... q_{n}$ are natural numbers.

Thus, every composite number can be expressed as a product of primes apart from the order.

Example:

Consider a composite number

$26950$ .

Let us factor this number using the factor tree method.

The prime factor of

$26950$

$=$

$2 \times 5 \times 5 \times 7 \times 7 \times 11$ .

That is,

$26950 = 2 \times 5^2 \times 7^2 \times 11$ .

Here, a composite number

$26950$ is written as a product of prime numbers.

If we change the order of the prime numbers, the answer will also be the same composite number.

We can write

$26950 = 2 \times 7^2 \times 5^2 \times 11$ or

$26950 = 11 \times 7^2 \times 5^2 \times 2$ .

Thus, the prime factorization of a natural number is unique, except for the order of its factors.

Important!

Recall:

HCF

$=$ Product of the smallest power of each common prime factor in the numbers.

LCM

$=$ Product of the greatest power of each prime factor involved in the numbers.

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1. The fundamental theorem of arithmetic

Theory: