Every composite number can be expressed as a product of primes, and this expression is unique, apart from the order in which they appear.
Applications:

To locate HCF and LCM of two or more positive integers.
To prove irrationality of numbers.
To determine the nature of the decimal expansion of rational numbers.
Algorithm to locate HCF and LCM of two or more positive integers:

Step I:
Factorize each of the given positive integers and express them as a product of powers of primes in ascending order of magnitude of primes.
Step II:
To find HCF, identify common prime factor and find the least powers and multiply them to get HCF.
Step III:
To find LCM, find the greatest exponent and then multiply them to get the LCM.

To prove Irrationality of numbers:

The sum or difference of a rational and an irrational number is irrational.
The product or quotient of a non-zero rational number and an irrational number is irrational.

To determine the nature of the decimal expansion of rational numbers:

Let x = $\dfrac{p}{q}$ , p and q are co-primes, be a rational number whose decimal expansion terminates. Then the prime factorization of ’q’ is of the form ${2^m}{5^n}$ , m and n are non-negative integers.
Let x = $\dfrac{p}{q}$ be a rational number such that the prime factorization of ‘q’ is not of the form ${2^m}{5^n}$ , ‘m’ and ‘n’ being non-negative integers, then x has a non-terminating repeating decimal expansion.

Examples

Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, and 15 respectively.
Solution:

$398 - 7 = 391,436 - 11 = 425,542 - 15 = 527$
HCF of $391,425,527 = 17$

Express 98 as a product of its primes.
Solution:
$2 \times {7^2}$

If the HCF of 408 and 1032 is expressible in the form $1032 \times 2 + 408 \times p$ , then find the value of p.
Solution:
HCF of 408 and 1032 is 24.

$\begin{array}{l}1032 \times 2 + 408 \times (p) = 24\\408p = 24 - 2064\\p = - 5\end{array}$

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