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LCM

Least Common Multiple (LCM)

To find the LCM of given numbers, following two methods are used (প্ৰদত্ত সংখ্যাৰ LCM বিচাৰিবলৈ তলত দিয়া দুটা পদ্ধতি ব্যৱহাৰ কৰা হয়)

Prime Factorisation Method
Division Method

1. Prime Factorisation Method

Step I : Break the given numbers into their prime factors (প্ৰদত্ত সংখ্যাবোৰক সিহঁতৰ মৌলিক গুণকত বিভক্ত কৰা)
Step II : Find the product of highest powers of all the factors which occur in the given numbers (প্ৰদত্ত সংখ্যাত দেখা দিয়া সকলো গুণকৰ সৰ্বোচ্চ শক্তিৰ গুণফল বিচাৰক।)
Step III : This product is the required LCM (এই প্ৰডাক্টটোৱেই হৈছে প্ৰয়োজনীয় LCM)

e.g. If we are to determine to LCM of 6, 8 and 12

6 = 2 x 3 = 2¹ x 3¹

8 = 2 x 2 x 2 = 2³

12 = 2 x 2 x 3 = 2² x 3¹

∴ Required LCM = 2³ x 3 = 8 x 3 = 24

2. Division Method

In this method, following steps are applied to find the LCM of the given numbers (এই পদ্ধতিত প্ৰদত্ত সংখ্যাসমূহৰ LCM বিচাৰিবলৈ তলত দিয়া পদক্ষেপসমূহ প্ৰয়োগ কৰা হয়)

List the Numbers : Write down the given numbers in a line, separated by commas.
Divide by a Common Prime : Divide by any prime number that exactly divides at least two of the given numbers
List the Quotients and Remaining Numbers : Write down the quotients and the undivided numbers in a line below the first
Repeat : Continue this process until the numbers in the line are relatively prime (no common divisors other than 1)
Calculate the LCM : Multiply all the divisors used and the numbers in the final line to get the required LCM

e.g. LCM of 16, 24, 36 and 54 is determined (16, 24, 36 और 54 का LCM निर्धारित किया गया है)

∴ Required LCM = 2 x 2 x 3 x 3 x 2 x 2 x 3 = 432

HCF and LCM of Fractions

$\mathbf{\textrm{LCM of Fractions}=\frac{\textrm{LCM of Numerators}}{\textrm{HCF of Denominators}}}$

$\mathbf{\textrm{HCF of Fractions}=\frac{\textrm{HCF of Numerators}}{\textrm{LCM of Denominators}}}$

$\mathbf{\textrm{Find the LCM of 1/2,3/2 and 9/4}}$

$=\frac{1}{2},\frac{3}{2}and\frac{9}{4}$

$=\frac{LCM of 1,3,9}{HCF of 2,2,4}$

$=\frac{9}{2}$

The product of LCM and HCF of 3/5 and 8/3.

HCF and LCM of Decimal Numbers

Steps to Find HCF/LCM of Decimal Numbers

Equalize the Decimal Places : Ensure all the decimal numbers have the same number of decimal places by adding zeros to the end of the numbers if necessary.
Remove the Decimal Points : Convert the decimal numbers to integers by removing the decimal points.
Find the HCF or LCM : Calculate the HCF or LCM of these integers as you would normally do.
Adjust for Decimal Places : Convert the result back to a decimal by placing the decimal point in the correct position.

Relation between LCM and HCF

If A and B are two numbers whose LCM and HCF are L and H, respectively, then

A x B = L x H

Product of numbers = Product of LCM and HCF