Formulas you need to remember :
1. when you want to Calculate the least Number which when gets divided by numbers namely x,yand z and leaves remainders as a,b and c respectively, then the number can be calculated as ,
number = [LCM of (x,y,z)-p], where p=(x-a)=(y-b)=(z-c)
2.when you want to calculate the least number which when divided by x,y and z leaves the same remainder in each case , then the number will be,
number =[LCM of (x,y,z)+R], where R is the remainder left in each case
3. LCM(a,b) * HCF(a,b) = product of the two numbers that is( a*b)
=>HCF = (a*b)/LCM
4. LCM and HCF of fractions :
LCM of fractions = LCM of numerators/ HCF of denominators
e.g. LCM of (5/7,6/5) = LCM(5,6)/HCF(7,5)
HCF of fractions = HCF of numerators/ LCM of denominators
e.g. HCF of (3/7,4/5) = HCF (3,4)/LCM(7,5)
5. when you want to calculate the greatest number that will divide x,y and z leaving the remainders as a,b and c , respectively then the number will be the , HCF ((x-a),(y-b),(z-c))
6. when you want to calculate the greatest number that will divide numbers x,y and z leaving the same remainder in all the cases , the number will be equal to the , HCF (|x-y|,|y-z|,|z-x|)
,where |x| is the modulus operation .
HOW TO CALCULATE HCF AND LCM :
There are two method for calculating the lcm of two numbers _ (1) Prime Factorisation Method (2) LEAST COMMON MULTIPLE Factorisation Method
→ Prime factorisation for calculating lcm
In this method we write down the prime factors of the numbers , then the LCM is the product of the highest powers of all the factors.
e.g Q. Calculate the LCM of 25, 30, 40.
so the factors are ,
25 = 5*5=(5²) , 30= 5*2*3, 40=5*2*2*2=5*(2³), where bª means b to the power of a
LCM of 25,30,40 = (5²)*(2³)*(3¹)= 600
→ Least Common Multiple of more than two numbers by Factorisation
in this method you divide all the numbers or as many as possible by such a prime common divisors as may be contained in them and then you multiply the divisors together and the final quotients.
e.g. Q. Calculate the LCM of 15 , 14 and 20.
5 | 15,14,20
2 | 3,14,4
7 | 3,7,2
Now , LCM (15,14,20)= 5*2*7*3*2=420
HCF can be calculated by using two methods _ (1) Common Factor Method (2) Highest Common Factor
→ Common Factor : a common factor of two or more than two numbers which divides each of them is calculated
HCF of (5,3,2)= 1
HCF of (3,6,9)=3
HCF of (25,30,40)= 5
HCF of 1445, 1190 =
so , the HCF is 85 .
QUESTIONS TO PRACTICE :
|HCF AND LCM QUESTIONS 1|
|HCF AND LCM QUESTIONS 2|
|HINTS AND SOLUTIONS 1|
|HINTS AND SOLUTIONS 2|
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