LCM & HCF
Factors:-
If a number ‘a’ divides another number ‘b’ exactly then ‘a’ is called a factor of b.
Multiple:-
If a number ‘b’ is exactly divisible by another number ‘a’ then ‘b’ is called multiple of a.
Ex:
Consier two numbers 6 & 24
6 is a factor of 24
& 24 is a multiple of 6.
HCF or GCD:-
The HCF of two or more than two numbers is the greatest number that divides each of them exactly.
Ex:- Find the HCF of 24, 36, 60
The highest number which divides these numbers exactly is 12.
∴ HCF ⟶ 12 Answer
There are three methods to find HCF of given set of numbers
(i) Factorization Method
(ii) Division Method
● Rules of divisibility are very helpful in finding the HCF or finding the answer using given options in competitive exams. So learn them properly.
(i) Factorization Method:-
Express each of the given numbers as the product of prime factors. Product of least powers of common prime factors gives HCF.
∴ H.C.F = 2³ = 8 Answer
Ex:- Find HCF of 96, 232
Solution:-
∴ HCF = 2³ = 8 Answer
Ex:- Find the HCF of 390, 1001
Sol:-
∴ HCF = 13 Answer
Ex:- Find the HCF of 121200, 65824, 19008
Sol:-
∴ HCF = 2⁵ = 32 Answer
Division Method:-
Suppose we have to find the HCF of two given numbers. Divide the larger number by the smaller one. Now divide the divisor by the remainder. Repeat this recursive process of dividing the preceding by the remainder last obtained till zero is obtained as the remainder. The last divisor is the required HCF.
Finding the HCF of more than two number: Suppose we are to find the HCF of three numbers then HCF of [(HCF of any two numbers) and (the third number0] gives the HCF of three given numbers.
Similarly, the HCF of more than three numbers may be obtained.
Ex:- Find the HCF of 24, 64
Sol:-
Ex:- Find the HCF of 96, 132
Sol:-
Ex:- Find the HCF of 390, 1001
Sol:-
Ex:- Find the HCF of 1512000, 65824, 19008
Sol:-
now HCF of 32, 19008
∴ HCF of three numbers is 32 Answer
(iii) Difference Method:-
From the given set of numbers take those two numbers whose difference is least. Let this difference be d.
The HCF is either ‘d’ or a number less than d which is a multiple of ‘d’, which divides given numbers exactly.
Ex:- Find HCF of 24, 64
Sol:-
Ex:- Find HCF of 96, 232
Sol:-
Ex:- Find HCF of 390, 1001
Sol:-
Ex:- Find HCF of 1512000, 65824, 19008
Sol:-
by observation, given numbers are multiple of 8 so HCF will also be multiple of 8, move further & we find HCF = 8×2×2 = 32 Answer
Ex:- Find HCF of 184, 230, 276
Sol:-
Ex:- Find the HCF of 216, 423, 1215, 1422, 2169, 2223
Sol:-
Take two numbers which have least difference.
Since sum of digits of each number is divisible by 9 so each number will also be divisible by 9.
Ex:- Find HCF of 960 and 1020
Sol:-
● The HCF of two or more than two numbers is smaller than or equal to the smallest number of given numbers
● The greatest number which divides a,b and c to leave the remainder R is HCF of (a – R), (b – R) and (c – R)
● The greatest number which divides x,y,z to leave remainders a,b and c respectively is HCF of (x – a), (y – b) and (z – c)
● The HCF of two or moe prime numbers will always be 1.
● HCF of co-prime numbers is 1
● HCF of fraction = \(\frac{\text{HCF of Numerator}}{\text{LCM of Denominator}}\)
Ex:- Find the HCF of \(\frac{{12}}{{25}}\), \(\frac{{9}}{{10}}\), \(\frac{{18}}{{35}}\), \(\frac{{21}}{{40}}\)
Sol:-
LCM:-
The least number which is exactly divisible by each one of the given numbers is called their LCM.
LCM ⟶ Least common multiple
Ex:- Find LCM of 25, 10, 35, 40
Sol:-
Ex:- Find LCM of 16, 20
Sol:-
Ex:- Find LCM of 10, 18, 20
Sol:-
● LCM by division method:-
⟶ First, write the numbers and separate them with commas
⟶Now divide the numbers by the smallest prime number
⟶ If any number is not divisible then write down that number and proceed further
⟶ Keep on dividing the row of numbers by prime numbers unless we get the results as 1 in the complete row
⟶ Now LCM of the numbers will be equal to the product of all the prime numbers we obtained in the division method
Ex:- LCM of 10, 18, 20
Sol:-
∴ LCM = 2 ×2 × 3 × 3 × 5 = 180 Answer
Ex:- LCM of 14, 20, 30, 21
Sol:-
∴ LCM = 2 × 2 × 3 × 5 × 7 = 420 Answer
Factor Method
● The LCM is always greater than or equal to the larger of the given numbers
● If two numbers are co-prime, their LCM is equal to their product
● The LCM of two or more numbers is divisible by each of the given numbers
● If LCM is divisible by a number A then it will also be divisible by all the factors of number A.
● LCM of two distinct prime numbers is their product
● The HCF of (Sum of Numbers and LCM of Numbers) will also be the HCF of these numbers.
Ex:- The sum and LCM of two numbers are 204 & 864. Find the numbers.
Sol:-
HCF of(204, 864) = HCF of numbers
∴ HCF of numbers will be 12
Let numbers are 12a & 12b where a & b are co-prime.
12a + 12b = 204
a + b = 17 ……………..(i)
& LCM = 12ab = 864 ⟹ ab = 72 …………….(ii)
a = 8 b = 9
∴ Numbers are :
12 × 8 = 96
12 × 9 = 108 Answer
● Let two numbers a 7 b then
Product of two numbers = HCF × LCM
● Product of ‘n’ numbers = (HCF)ⁿ⁻¹ × LCM
Ex:- The LCM of 5 numbers is 240 and their HCF is 3. Find the multiplication of all the numbers
Sol:-
multiplication of all numbers = (HCF)⁵⁻¹ × LCM
= (3)⁴ × 240
= 19440 Answer
● LCM of fraction = \(\frac{{LCM\;of\;Numerator}}{{HCF\;of\;Denominator}}\)
= \(\frac{{LCMof(12,\;20,\;18)}}{{HCFof(25,\;27,\;25)}}\)
= \(\frac{{180}}{1}\)
= 180 Answer
Ex:- Find LCM of 2.4, 0.36 and 7.2
Sol:-
To calculate the LCM of decimal integers, make the decimal integers into whole numbers multiplying with any friendly numbers i.e. 10, 100, 1000, ………. etc.
Now find out LCM of these whole numbers Finally divide this LCM by that friendly number which we used earlier.
∴ LCM = 2⁴ × 3² × 5 = 720
∴ required LCM = \(\frac{{720}}{{100}}\) = 7.2 Answer
Ex:- Find the LCM of 48, 168, 324 and 1400
Sol:-
∴ LCM = 2⁴ × 3⁴ × 5² × 7
= 226800 Answer
Method (2):
∴ LCM = 2⁴ × 3⁴ × 5² × 7 = 226800
Ex:- Find the HCF of 48, 168, 324, 1400
Sol:-
Method (1):
2² is common in all
∴ HCF=2²=4
Method (2):
Ex:- Find the least number which when divided by 48, 18, 42, 72 leaves the remainder 41, 11, 35, 65 respectively.
Sol:-
In this tye of question the difference between the divisor & the corresponding remainder is same in each case
48 – 41 = 7, 18 – 11 = 7, 42 – 35 = 7, 72 – 65 = 7
∴ required number = LCM of (48, 18, 42, 72) – 5
∴ LCM = 2⁴ × 3² × 7 = 1008
∴ required number = 1008 – 7 = 1001 Answer
Ex:- Find the greatest number which is such that when 12288, 19139 and 28200 are divided by it, the remainder are all the same.
Sol:-
In this type of question, first find out the difference of these numbers & then find HCF of the numbers found out in differences
● Concept:- If two numbers have the same remainder when divided by a divisor, then their difference is divisible by that divisor.