Basic Concept of LCM and HCF with Examples

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Basic Concept of LCM and HCF with Examples

Basic Concept of LCM and HCF with Examples

Hi Aspirants,
Today we are presenting you the basic concepts of Least Common Multiple (LCM) and Highest Common Factor (HCF) with examples. You can expect 1-2 questions in your upcoming competitive examinations. Lets start.

What is Factor

One number is said to be a factor of the another when it divides the other exactly. Thus, 6 and 7 are factors of 42.

What is Common Factor

A common factor of two or more numbers is a number that divides each of them exactly. Thus, 3 is a common factor of 9, 18, 21 and 33.

Least Common Multiple (LCM)

What is L.C.M?

The least number which is exactly divisible by each one of the given numbers is called their Least Common Multiple (LCM).

Methods to Find L.C.M of a Given Set of Numbers

There are two methods of finding the L.C.M. of a given set of numbers:

I. Prime Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

II. Division Method (short-cut): Arrange the given numbers in a row in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1 The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

Example 1 – Finding the Numbers Given their LCM

Question
The L.C.M. of two numbers is 48. The numbers are in the ratio 2 : 3. Then sum of the number is:

Answer
Let the numbers be 2x and 3x.
Then, their L.C.M. =6x.
As per the question 6x=48 or x=8.
The numbers are 16 and 24.
Hence, required sum =(16+24)=40.

Example 2 – LCM of Fractions

Question
Find the LCM of \frac{3}{5}, \frac{21}{25} and \frac{14}{15}

Answer
Step 1: Find the LCM of the numerator of 3,21,14=42
Step 2: HCF of the denominator of 5,25,15=5
Required LCM of given fractions =42/5=8(2/5)

Highest Common Factor (H.C.F.)

What is H.C.F?

Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.). The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.

Methods to Find H.C.F of Two Numbers

There are two methods of finding the H.C.F. of a given set of numbers:

I. Prime Factorization Method: Express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.

II. Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor which makes reminder zero is the required H.C.F.

Methods to Find H.C.F of Three or More Numbers

Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of the three given number.

Similarly, the H.C.F. of more than three numbers may be obtained.

Example 1 – Factors and Multiples

Question
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

Answer
Calculate the H.C.F. of the differences between all the numbers.

As the number required is the Highest Common Factor of the difference which will divide the difference exactly And hence, whatever left is the remainder which is equal for each of the number to be divided.

Required number = H.C.F. of (91 – 43), (183 – 91) and (183 – 43)
= H.C.F. of 48, 92 and 140
= 4.

Example 2 – Factors and Multiples

Question
Let M be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then find sum of the digits in M .

Answer
Required number M = H.C.F. of (4665 – 1305), (6905 – 4665) and (6905 – 1305)
= H.C.F. of 3360, 2240 and 5600
= 1120.
Sum of digits in M = (1+1+2+0)
= 4

Example 3 – HCF of Fractions

Question
Find HCF of \frac{4}{5}, \frac{5}{6} and \frac{9}{10}

Answer
Step 1: Find HCF of the numerators 4,5,9=1
Step 2: Find LCM of denominator 5,6,10=30
HCF of given fractions =\frac{1}{30}

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