You all know that speed in calculation sets the complete base for Quantitative Aptitude section of various competitive exams and if you know enough Short Tricks in Quant Section, you will surely score better in the section. So, let us make it easy for all of you through these Simple and Easy Concepts to solve LCM & HCF Problems which will not only make quant questions easy but will also save your time. The tricks will be helpful for the upcoming Indian Railways recruitment (RRB) Exam and SSC CGL 2016 Exam.
HCF & LCM are acronym for words, Highest common factor and Lowest common multiple respectively.
For example:
(1) 24 = 2 x 2 x 2 x 3 = 2^3 x 3.(2) 420 = 2 x 2 x 3 x 5 x 7 = 2^2 x 3 x 5 x 7
The Least Common Multiple of two or more integers is always divisible by all the integers it is derived from. For example, 20 is a multiple of 5 because 5 × 4 = 20, so 20 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 4.
LCM can also be understand by this example:
Multiples of 5 are:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70 …
And the multiples of 6 are:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, …
Common multiples of 5 and 6 are:
30, 60, 90, 120, ….
Hence, the lowest common multiple is simply the first number in the common multiple list i.e 30.
By Prime Factorizations
The prime factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers.
Example: To find the value of LCM (9, 48, and 21).
First, find the factor of each number and express it as a product of prime number powers.
Like 9 = 32,
48 = 24 * 3
21 = 3 * 7
Then, write all the factors with their highest power like 32, 24, and 7. And multiply them to get their LCM.
Hence, LCM (9, 21, and 48) is 32 * 24 * 7 = 1008.
By Division Method
Here, divide all the integers by a common number until no two numbers are further divisible. Then multiply the common divisor and the remaining number to get the LCM.
2 | 72, 240, 196
2 | 36, 120, 98
2 | 18, 60, 49
3 | 9, 30, 49
| 3, 10, 49
L.C.M. of the given numbers
= product of divisors and the remaining numbers
= 2×2×2×3×3×10×49
= 72×10×49 = 35280.
H.C.F. of 72 and 126 = 18
The product of L.C.M. and H.C.F. of two natural numbers = the product of the numbers.
For Example:
LCM (8, 28) = 56 & HCF (8, 28) = 4
Now, 8 * 28 = 224 and also, 56 * 4 = 224
Formulae for finding the HCF & LCM of a fractional number.
HCF of fraction = HCF of numerator / LCM of denominator
LCM of Fraction = LCM of Numerator / HCF of Denominator
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