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.
