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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 = , p and q are co-primes, be a rational number whose decimal expansion terminates. Then the prime factorization of ’q’ is of the form, m and n are non-negative integers.
- Let x = be a rational number such that the prime factorization of ‘q’ is not of the form , ‘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*:

HCF of

- Express 98 as a product of its primes.

*Solution*:

- If the HCF of 408 and 1032 is expressible in the form, then find the value of p.

*Solution*:

HCF of 408 and 1032 is 24.

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