LCM and HCF
Factor:- Factor is a number which exactly divides the another number.
As 2,5 are factor of 10.
Multiple:- A number is said to be multiple of another number, when it is exactly
divisible by other number.
As 10 is multiple of 2,5.
LCM :- Least common multiple is a number which is smallest multiple of two or more
than two numbers.
For example:- The common multiples of 3 and 4 are 12,24,36 and so on and
the least (smallest) multiple is 12.Hence LCM of 3 and 4 is 12.
For example:- The multiples of 4 are
(1) Prime factorization method
(i) Express the given number s product of prime numbers.
(ii) Find each prime factor and their highest index.
(iii) The product of all prime factor with highest index is LCM of given numbers.
Example:-LCM of 12,14,16,18.
12=22*3
14=2*7
16=24
18=2*32
LCM=24*32*7=1008
(2) Division Method
(i) Draw a table as shown in below.
(ii) Divides the numbers with their common factors.
(iii) Divides till the numbers have no common factors.
(iv) At last multiply common factor and last line quotients.
Example:-LCM of 12,14,16,18.
LCM=2*2*3*1*7*4*3=1008
HCF & GCD :- Highest common factor is the greatest divisor of two or more than two numbers.
For example:- The common divisors of 8 and 12 are 1,2,4. and the greatest common
divisor is 4.
For example:- The factors of 24 are
(1) Prime factorization method
(i) Express the given number s product of prime numbers.
(ii) Find each prime factor and their lowest index.
(iii) The product of all prime factor with lowest index is HCF of given numbers.
Example:-HCF of 12,16,
12=22*3
16=24
HCF=22*30=4.
(2) Division method
(i) Divide the largest given number by second numbers.
(ii) Divide the second number by above remainder of (i).
(iii) Divide the remainder of (i) by remainder of (ii).Continue the process
till the remainder is not zero
Example:-HCF of 12,16,
12⎾16⏋1
12
__
4⎾12⏋3
12
__
0
HCF=4.
Some Important Formula :-
(1) Product of two numbers = Their HCF. * Their LCM
(2) HCF of given fractions = HCF of numerators / LCM of Denominators
(3) LCM of given fractions =LCM of numerators / HCF of Denominators
Factor:- Factor is a number which exactly divides the another number.
As 2,5 are factor of 10.
Multiple:- A number is said to be multiple of another number, when it is exactly
divisible by other number.
As 10 is multiple of 2,5.
LCM :- Least common multiple is a number which is smallest multiple of two or more
than two numbers.
For example:- The common multiples of 3 and 4 are 12,24,36 and so on and
the least (smallest) multiple is 12.Hence LCM of 3 and 4 is 12.
For example:- The multiples of 4 are
4,
8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, . . .
The multiples of 6 are
6,
12, 18, 24, 30, 36, 42, 48, 54, 60, 66, . . .
The common multiples of 4 and 6 are
12,
24, 36, 48, . . .
The Least Common Multiples of 4 and 6 is
12
LCM(4, 6) = 12
Methods to find LCM(1) Prime factorization method
(i) Express the given number s product of prime numbers.
(ii) Find each prime factor and their highest index.
(iii) The product of all prime factor with highest index is LCM of given numbers.
Example:-LCM of 12,14,16,18.
12=22*3
14=2*7
16=24
18=2*32
LCM=24*32*7=1008
(2) Division Method
(i) Draw a table as shown in below.
(ii) Divides the numbers with their common factors.
(iii) Divides till the numbers have no common factors.
(iv) At last multiply common factor and last line quotients.
Example:-LCM of 12,14,16,18.
2
|
12
|
14
|
16
|
18
|
2
|
6
|
7
|
8
|
9
|
3
|
3
|
7
|
4
|
9
|
1
|
7
|
4
|
3
|
For example:- The common divisors of 8 and 12 are 1,2,4. and the greatest common
divisor is 4.
For example:- The factors of 24 are
1, 2, 3, 4, 6, 8, 12,
24
The factors of 36 are
1, 2, 3, 4, 6, 9, 12,
18, 36
The common factors of
24 and 36 are
1, 2, 3, 4, 6, 12
The Greatest Common Divisor of 24 and 36 is 12
GCD(24, 36) = 12
Methods to find HCF(1) Prime factorization method
(i) Express the given number s product of prime numbers.
(ii) Find each prime factor and their lowest index.
(iii) The product of all prime factor with lowest index is HCF of given numbers.
Example:-HCF of 12,16,
12=22*3
16=24
HCF=22*30=4.
(2) Division method
(i) Divide the largest given number by second numbers.
(ii) Divide the second number by above remainder of (i).
(iii) Divide the remainder of (i) by remainder of (ii).Continue the process
till the remainder is not zero
Example:-HCF of 12,16,
12⎾16⏋1
12
__
4⎾12⏋3
12
__
0
HCF=4.
Some Important Formula :-
(1) Product of two numbers = Their HCF. * Their LCM
(2) HCF of given fractions = HCF of numerators / LCM of Denominators
(3) LCM of given fractions =LCM of numerators / HCF of Denominators
Some Important Results:-
(1) Largest number which divides x,y,z to leave same remainder = HCF of y-x, z-y, z-x.
(2) Largest number which divides x,y,z to leave same remainder
R =HCF of x-R, y-R, z-R.
R =HCF of x-R, y-R, z-R.
(3) Largest number which divides x,y,z to leave same remainder a,b,c
=HCF of x-a, y-b, z-c.
=HCF of x-a, y-b, z-c.
(4) Least number which when divided by x,y,z and leaves a remainder R in each case
= ( LCM of x,y,z) + R
Example 1:-Find the HCF and LCM of the numbers 60 and 72.
Example 3:- Least number which when divided by 35,45,55 and leaves remainder
18,28,38; is?
Example 1:-Find the HCF and LCM of the numbers 60 and 72.
write each number as a
product of its prime factors.
Explanation. Now 60 = 2 * 2 * 3 * 5= 22*3 * 5
|
72 = 2 * 2 *2* 3*3 = 23*32
HCF=22*31=12.
LCM=23*32* 5=360.
Example 2:-Find the LCM of 2/3 and 4/6.
Explanation. LCM of 2 and 4 is 4.
HCF of 3 and 6 is 3.
Formula
LCM of given fractions =LCM of numerators / HCF of Denominators.
Hence LCM of 2/3 and 4/6 is 4/3.
Ans is 4/3.
|
18,28,38; is?
Explanation. Here the difference 35-18=17,45-28=17 55-38=17
Here the difference between every divisor and remainder is same i.e. 17.
Therefore, required number = LCM of (35,45,55)-17 =3465-17= 3448.
Example 4:-Find the least number which when divided by 5,7,9 and 12,
leaves the
same remainder 3 in each case.
If one number is 60.Find the other number.
Explanation. Formula
Product of two numbers = Their HCF. * Their LCM
60 * Other number =30*300
Other number = 30*300/60 =150.
same remainder 3 in each case.
Explanation. In these type of questions, we need to find the LCM of the
divisors and
add the common remainder (3) to it.
So, LCM (5, 7, 9, 12) = 1260
Therefore, required number = 1260 + 3 = 1263
Example 5:-The LCM and HCF of two positive numbers are 300
and 30 respectively.add the common remainder (3) to it.
So, LCM (5, 7, 9, 12) = 1260
Therefore, required number = 1260 + 3 = 1263
If one number is 60.Find the other number.
Explanation. Formula
Product of two numbers = Their HCF. * Their LCM
60 * Other number =30*300
Other number = 30*300/60 =150.
Ans is 150.
Example 6:-Two numbers are in the ratio of 5:11. If their HCF is 7, find the numbers and
their LCM.
Explanation. Let the numbers be 5k and 11k. Since 5:11 is already the reduced ratio,
since they have no common factor hence ‘k’ has to be the HCF.
So, the numbers are 5 x 7 = 35 and 11 x 7 = 77.
LCM = 35*77/7 = 385.
Example 7:-Two numbers are in the ratio 2:3. If the product of their LCM and HCF is 294,
find the numbers.
Explanation.Let the common ratio be ‘k’. So, the numbers are 2k and 3k.
Formula
Product of two numbers = Their HCF. * Their LCM
2k x 3k = 294
Example 6:-Two numbers are in the ratio of 5:11. If their HCF is 7, find the numbers and
their LCM.
Explanation. Let the numbers be 5k and 11k. Since 5:11 is already the reduced ratio,
since they have no common factor hence ‘k’ has to be the HCF.
So, the numbers are 5 x 7 = 35 and 11 x 7 = 77.
LCM = 35*77/7 = 385.
Example 7:-Two numbers are in the ratio 2:3. If the product of their LCM and HCF is 294,
find the numbers.
Explanation.Let the common ratio be ‘k’. So, the numbers are 2k and 3k.
Formula
Product of two numbers = Their HCF. * Their LCM
2k x 3k = 294
k2 = 49
k = 7
LCM = 35*77/7 = 385.
Example 8:-Find the LCM and HCF of 0.70,1.75 and 5.6.
Explanation. The given numbers can be written as 0.70,1.75 and 5.60
Without decimal pint numbers are 70,175 and 560
LCM of 70,175 and 560 is 1400.
Hence LCM of 0.70,1.75 and 5.6 is 14
HCF of 70,175 and 560 is 35.
Hence HCF of 0.70,1.75 and 5.6 is 0.35
Example 9:-Find the greatest number which on dividing 70 and 50 leaves remainders
1 and 4 respectively.
Explanation.The required number leaves remainders 1 and 4 on dividing 70 and 50
respectively.
This means that the number exactly divides 69 and 46.
So, we need to find the HCF of 69 and 46.
HCF of 69 and 46 is 23
k = 7
LCM = 35*77/7 = 385.
Example 8:-Find the LCM and HCF of 0.70,1.75 and 5.6.
Explanation. The given numbers can be written as 0.70,1.75 and 5.60
Without decimal pint numbers are 70,175 and 560
LCM of 70,175 and 560 is 1400.
Hence LCM of 0.70,1.75 and 5.6 is 14
HCF of 70,175 and 560 is 35.
Hence HCF of 0.70,1.75 and 5.6 is 0.35
Example 9:-Find the greatest number which on dividing 70 and 50 leaves remainders
1 and 4 respectively.
Explanation.The required number leaves remainders 1 and 4 on dividing 70 and 50
respectively.
This means that the number exactly divides 69 and 46.
So, we need to find the HCF of 69 and 46.
HCF of 69 and 46 is 23
Ans is 23.
Example 10:-Find Least number which when divided by 5,6,7,8 and leaves remainder 3,
but when divided by 9, leaves no remainder.
Explanation. LCM of 5,6,7,8 = 840
Required number = 840 k + 3
Least value of k for which (840 k + 3) is divided by 9 is 2
Therefore, required number = 840*2 + 3
= 1683
but when divided by 9, leaves no remainder.
Explanation. LCM of 5,6,7,8 = 840
Required number = 840 k + 3
Least value of k for which (840 k + 3) is divided by 9 is 2
Therefore, required number = 840*2 + 3
= 1683
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