Number System Tutorial II: Divisibility, Remainder, HCF & LCM

    1 Votes

LCM and HCF

LCM (Least common multiple) of a given set of quantities is the least possible quantity which can be divisible by each of the given quantities in the set.

HCF (Highest common factor)/ GCD (Greatest common divisor) of a given set of quantities is the highest possible quantity which can exactly divide each of the given quantities in the set.

Prime factorization method for finding LCM and HCF

Prime factorization is an easy method for LCM and HCF for a given group of numbers. Let's look at an example to see how to find LCM and HCF of given set of numbers using prime factorization method.
Illustrative example: Find the LCM and HCF of 96, 108 and 120?
In this method, first find the prime factors of the given numbers. Prime factorized expressions of the numbers are given below.
`96 = 2^5 xx 3`
`108 = 2^2 xx 3^3`
`120 = 2^3 xx 3 xx 5`
LCM = product of maximum exponent forms of each distinct bases which are occurred in any of the given numbers.
`:. "LCM" = 2^5 xx 3^3 xx 5 = 4320`
HCF = product of maximum exponent form of distinct bases which are occurred commonly in all the numbers.
`:. "HCF" = 2^2 xx 3 = 12`

 

Find the LCM and HCF of 48, 36 and 72.?
Prime factorized expressions of the numbers are given below.
`48 = 2^4 xx 3`
`36 = 2^2 xx 3^2`
`72 = 2^3 xx 3^2`
LCM = product of maximum exponent forms of each distinct bases which are occurred in any of the given numbers.
`:. "LCM" = 2^4 xx 3^2 = 144`
HCF = product of maximum exponent form of distinct bases which are occurred commonly in all the numbers.
`:. "HCF" = 2^2 xx 3 = 12`

LCM and HCF of fractions

`"LCM of fractions" = "LCM of numerators"/"HCF of denominators"`
Find the LCM of `6/5`, `9/10` and `27/20`?
LCM of `6/5`, `9/10` and `27/20` `= "LCM(6, 9, 27)"/"HCF(5, 10, 20)" = 54/4`
`"HCF of fractions" = "HCF of numerators"/"LCM of denominators"`
Find the HCF of `16/15`, `20/25` and `36/35`?
HCF of `16/15`, `20/25` and `36/35` `= "HCF(16, 20, 36)"/"LCM(15, 25, 35)" = 4/525`

Key points on LCM and HCF

  • Product of any two numbers is equal to the product of their LCM and HCF.
    Eg: Let us consider two numbers 12 and 8.
    LCM (12, 8) = 24
    HCF (12, 8) = 4
    `12 xx 8 = 24 xx 4`
    i.e. product of 12 and 8 = LCM(12, 8) xx HCF (12, 8)
  • HCF of any two numbers is a factor of the difference between the numbers.
    Eg: Let A = 36 and B = 63
    B - A = 63 - 36 = 27
    HCF (36, 63) = 9
    9 is a factor of 27.
  • Let A and B are two numbers and their LCM and HCF are 'L' and 'H' respectively. A and B can be expressed in terms of their HCF in the following manner.
    A = Hp and B = Hq, where p and q are two relatively prime numbers.
    LCM of A and B can be expressed in the form; LCM (A, B) = Hpq

    Eg: Let A = 48 and B = 32.
    LCM = 96 and HCF = 16
    A = 16 x 3, B = 16 x 2 : Here 3 and 2 are relatively prime numbers.
    `:. "HCF" = "16 and LCM" = 16 xx 2 xx 3 = 96`
Page 2 of 3

Popular Videos

communication

How to improve your Interview, Salary Negotiation, Communication & Presentation Skills.

Got a tip or Question?
Let us know

Related Articles

Numbers – Aptitude Test Questions, Tricks & Shortcuts
Boats and Streams - Aptitude Test Tricks, Formulas & Shortcuts
Mixtures & Alligations - Aptitude Test Tricks, Formulas & Concepts
Pipes and Cisterns - Aptitude Test Tricks, Formulas & Concepts
Time and Distance - Aptitude Test Tricks, Shortcuts & Formulas
Cyclicity of Numbers Aptitude Test Questions - Concepts Formulas Tips
Percentages - Aptitude Test Tricks & Shortcuts & Formulas
Time and Work - Aptitude Test Tricks & Shortcuts & Formulas
Problems on Trains - Aptitude Tricks & Shortcuts & Formulas
Number System Tutorial I: Integers, Fractions, Prime & Composite
Number System Tutorial Part III: Factors, Multiples, Unit Digit and Last Two Digits of Exponents
Time & Distance Tutorial I: Basic Concepts, Average Speed & Variation of Parameters
Time and Distance Tutorial II: Relative Speed, Linear & Circular Races, Meeting Points
Time and Distance Tutorial III: Boats & Streams, Escalators, Journey with Stoppages
Logarithm - Aptitude Questions, Rules, Formulas & Tricks
Probability - Aptitude Tricks, Formulas & Shortcuts
Permutations and Combinations - Tricks & important formulas
Sequence and Series - Important concepts, Formulas & Tricks
Ratio and Proportion - Important formulas, Shortcuts & Tricks
Quadratic Equations - Aptitude Tricks