Question

H.C.F And L.C.M Tutorials & Tricks

Answer 1

Introduction
LCM of a set of numbers would be greater than or equal to the largest number in that set.HCF of a given set of numbers would be smaller than or equal to the smallest number in that set.

Formulae & Examples
Formula for finding out H.C.F and L.C.M.
1. Factors and Multiples:
    If number a divided another number b exactly, we say that a is a factor of b.
    In this case, b is called a multiple of a.


2. The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.
   There are two methods of finding the H.C.F. of a given set of numbers:
    I. Factorization Method: Express the 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 is required H.C.F.
        Finding the H.C.F. of more than two 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 three given number.
        Similarly, the H.C.F. of more than three numbers may be obtained.


3.  Least Common Multiple (L.C.M.):The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
There are two methods of finding the L.C.M. of a given set of numbers:
I. 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 rwo 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.

4. Product of two numbers = Product of their H.C.F. and L.C.M.

               
5. Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.


6. H.C.F. and L.C.M. of Fractions:
    1. H.C.F. = H.C.F. of Numerators/L.C.M. of Denominators
    2. L.C.M. = L.C.M. of Numerators/H.C.F. of Denominators

               
7. H.C.F. and L.C.M. of Decimal Fractions: In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.


8. Comparison of Fractions: Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

9. HCF
    Common Factor A number is called to be a factor of another numbers when it divides other numbers exactly.

    eg. 4 is a common factor of 8, 12, 16, 20, 24


9. Highest Common Factor : -HCF of two or more numbers is the greatest number that divides each of them exactly.

10. Method of Prime Factors - Break the given numbers into prime factors and find the product of common prime factors, product will be HCF


Find the HCF of 42 and 70

    Solution- 42=2*3*7,   70=2*5*7 So HCF will be 2*7 =14

Find the HCF of 24, 45, 60

    Solution-  24 = 2*2*2*3
     45= 3*3*5
     60= 2*2*3*5   so HCF of 24, 45 and 60 is 3


11. Second Method = Wrting in a row and division by a common divisor of all-
    Step 1: Write the numbers in a row
    Step 2: Divide by a common divisor of all
    Step 3: Write the remainders in second row
    Step 4: Continue this process till we get all the remainders prme to one another:

Answer 2

Sample Example
Ex
Find the HCF of 16.5, 0.45 and 15 ?
A
These numbers can be written as 16.50, 0.45 and 15.00

    Now find the HCF of 1650, 45 and 1500 , we get HCF as 15, now convert to euivalent fraction whivh comes as 00.15  This is our required HCF
Ex
Find the HCF of 54/9, 3*(9/17), 36/51 ?
A
Express all fractions in their lowest terms6/1, 60/17, 12/17

    HCF=HCF of Numenerators/ LCM of Denominators= HCF of (6, 60, 12) / LCM of (1, 17, 17)= 6/17
Ex
Find the greatest number which will divide 410, 751 and 1030 so as to leave the remainde 7 in each case ?
A
Greatest number will be = HCF of (410-7), (751-7) and (1030-7)= HCF of 403, 744 and 1023
Ex
LCM of two co-prime numbers x and y where x>y is 161. Find out the value of 3y-x ?
A
HCF of x and y = 1, since they are co prime numbers

    Now, we know Product of two numbers = Product of their HCF and LCM

    So, xy = 1 * 161 = 161

    So co-primes can be ( 1, 161) or ( 23, 7 )

    Since x > y , so x= 23 and y = 7

    so, 3y - x = 3 * 7 - 23 = -2
Ex
The traffic lights at three different road crossings change after every 24, 36 and 48 secs respectively. If they all change simultaneously at 09 : 10 : 00 hours , then at what time will they againg change simultaneously ?
A
: Here time requirment is least so we will find LCM

    Change interval will be LCM of 24, 36, 48 = 144 sec

    144 sec= 2 min 24 sec, So interval of change will be at 09 : 12 : 24 hours
Ex
The least number, which when divided by 48, 60, 72, 108 and 140 leaves 38, 50, 62, 98 and 130 as remainders respectively, is : ?
A
Here (48 - 38 ) = ( 60 - 50 ) = ( 72 - 62 ) = ( 108 - 98 ) = ( 140 - 130 ) = 10 in every case

    Leat number will be LCM of ( 48, 60, 72, 108, 140 ) - 10

    LCM = 15120

    So, required number = 15120 - 10 = 15110