Number Theory

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Highest Common Factor (HCF)

HCF is the largest factor of two or more given numbers. The same can be defined algebraically as HCF of two or more algebraically expressions is the expression of highest dimension which divides each of them without remainder.

HCF is also called GCD (Greatest Common Divisor).

Product of two numbers = \(LCM \times HCF\)

LCM is a multiple of HCF

\({\rm{LCM}}\;{\rm{of}}\;{\rm{fractions}}\;{\rm{ = }}\cfrac{{{\rm{LCM}}\;{\rm{of}}\;{\rm{Numerators}}}}{{{\rm{HCF}}\;{\rm{of}}\;{\rm{denominators}}}}\)

\({\rm{HCF}}\;{\rm{of}}\;{\rm{fractions}}\;{\rm{ = }}\cfrac{{{\rm{HCF}}\;{\rm{of}}\;{\rm{Numerators}}}}{{{\rm{LCM}}\;{\rm{of}}\;{\rm{denominators}}}}\)

HCF of two numbers is same as

HCF( sum of numbers, LCM of numbers)

Example 01: If the HCF of \({2^3} \times {3^5} \times {7^4}\) and \(n\) is \({2^2} \times {3^5}\), then find the minimum value of \(n\)

The number \(n\) must contain 2 power of 2, at least 5 powers of 3 and does not contain any power of 7. So the minimum value of \(n\) is \({2^2} \times {3^5}\).

Example 02: The LCM of the numbers \(\frac{2}{3},\,\frac{4}{9},\,\frac{8}{{27}}\) is how many times of their HCF.

\({\rm{LCM}} = \frac{{{\rm{LCM}}\,\,{\rm{of}}\;(2,4,8)}}{{{\rm{HCF}}\;{\rm{of}}\;(3,9,27)}} = \frac{8}{3}\)

\({\rm{HCF}} = \frac{{{\rm{HCF}}\;{\rm{of}}\;{\rm{(2,}}\,{\rm{4,8)}}}}{{{\rm{LCM(3,9,27)}}}} = \frac{2}{{27}}\)

Hence \(\frac{{{\rm{LCM}}}}{{{\rm{HCF}}}} = \frac{{8/3}}{{2/27}} = 36\)..

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