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You all have studied how to find LCM and HCF in your LP-UP classes. But how many of you can still solve these kind of problems?There will be problems based on these in various competetive exams. (both direct and indirect). So do your best in this portion.
If a number 'x' divides another number 'y' exactly, then we say that 'x is a factor of 'y' and that 'y is a multiple of x'. For example, 5 is a factor of 10 and therefore 10 is a multiple of 5.
LCM is the least non-zero number in common multiples of two or more numbers.
Multiple of 4- 4, 8, 12, 16, 20, etc
Multiple of 5- 5, 10, 15, 20, 25, etc
Multiple of 6- 6, 12, 18, 24, 30, etc
Common multiples of 4 and 5 = 20, 40, etc
Least Common Multiple (LCM) = 20
Common multiples of 5 and 6 = 30, 60, etc
Least Common Multiple (LCM) = 30
2 32,64,40
2 16,32,20
2 8,16,10
2 4,8,5
2 2,4,5
1,2,5
.'.LCM = 2x2x2x2x2x2x 5=320
1 3,5,11
3,5,11
.'. LCM = 3 x 5 x 11 = 165
The least number divisible by 3,5 and 6 is their LCM, which is 30.
Now 30 = 2 x 3 x 5
3 3,5,6
1,5,2
To make it a perfect square, it must be multiplied by 2 x 3 x 5
.'.Required number = 30x2 x3x5 = 900
The Highest Common Factor of two or more numbers is the greatest number which divides each of them exactly.
For example, find the HCF of 21 and 27.
3 21,27
7,9
.'.HCF = 3
2 32,48,96
2 16,24,48
2 8,12,24
2 4,6,12
2,3,6
.'.HCF = 16
1 3,5,11
3,5,11
.'.HCF = 1
2 14,24,36
7,12,18
.'. HCF = 2
LCM of fractions = LCM of numerators/HCF of denominators
HCF of fractions = HCF of numerators/LCM of denominators
LCM of numerators = LCM of 1, 2, 3
= 1x2x3 = 6
HCF of denominators = HCF of 3, 5, 7
= 1
.'.LCM of the fraction =6/1=6
HCF of numerators = HCF of 3, 2,1
= 1
LCM of denominators = LCM of 4, 7, 5
= 140
.'.HCF of the fraction = 1/140
HCF of numerators = HCF of 12, 24, 32
2 12,24,32
2 6,12,16
3,6,8
.'.HCF of numerators = 2x2 = 4
LCM of denominators = LCM of 44, 20 and 10
2 44,20,10
2 22,10, 5
5 11,5,5
11, 1,1
LCM of denominators = 2x2x5x11 = 220
HCF of the fraction=4/220=1/55
LCM AND HCF
You all have studied how to find LCM and HCF in your LP-UP classes. But how many of you can still solve these kind of problems?There will be problems based on these in various competetive exams. (both direct and indirect). So do your best in this portion.
FACTORS AND MULTIPLES
If a number 'x' divides another number 'y' exactly, then we say that 'x is a factor of 'y' and that 'y is a multiple of x'. For example, 5 is a factor of 10 and therefore 10 is a multiple of 5.
LCM : LEAST COMMON MULTIPLE
LCM is the least non-zero number in common multiples of two or more numbers.
Multiple of 4- 4, 8, 12, 16, 20, etc
Multiple of 5- 5, 10, 15, 20, 25, etc
Multiple of 6- 6, 12, 18, 24, 30, etc
Common multiples of 4 and 5 = 20, 40, etc
Least Common Multiple (LCM) = 20
Common multiples of 5 and 6 = 30, 60, etc
Least Common Multiple (LCM) = 30
EXAMPLES
1.
Find the LCM of 32, 64, 402 32,64,40
2 16,32,20
2 8,16,10
2 4,8,5
2 2,4,5
1,2,5
.'.LCM = 2x2x2x2x2x2x 5=320
2.
What is the LCM of 3,5,111 3,5,11
3,5,11
.'. LCM = 3 x 5 x 11 = 165
3.
The least square number which is divisible by 3, 5 and 6 is___The least number divisible by 3,5 and 6 is their LCM, which is 30.
Now 30 = 2 x 3 x 5
3 3,5,6
1,5,2
To make it a perfect square, it must be multiplied by 2 x 3 x 5
.'.Required number = 30x2 x3x5 = 900
HCF; HIGHEST COMMON FACTOR
The Highest Common Factor of two or more numbers is the greatest number which divides each of them exactly.
For example, find the HCF of 21 and 27.
3 21,27
7,9
.'.HCF = 3
EXAMPLES
1.
Find the HCF of 32, 48, 962 32,48,96
2 16,24,48
2 8,12,24
2 4,6,12
2,3,6
.'.HCF = 16
2.
Find the HCF of 3, 5, 111 3,5,11
3,5,11
.'.HCF = 1
3.
Find the HCF of 14, 24, 362 14,24,36
7,12,18
.'. HCF = 2
TYPE-I
LCM of fractions = LCM of numerators/HCF of denominators
HCF of fractions = HCF of numerators/LCM of denominators
EXAMPLES
1.
Find the LCM of 1/3,2/5,3/7LCM of numerators = LCM of 1, 2, 3
= 1x2x3 = 6
HCF of denominators = HCF of 3, 5, 7
= 1
.'.LCM of the fraction =6/1=6
2.
Find the HCF of 3/4,2/7,1/5HCF of numerators = HCF of 3, 2,1
= 1
LCM of denominators = LCM of 4, 7, 5
= 140
.'.HCF of the fraction = 1/140
3.
Find the HCF of 12/44,24/20,32/10HCF of numerators = HCF of 12, 24, 32
2 12,24,32
2 6,12,16
3,6,8
.'.HCF of numerators = 2x2 = 4
LCM of denominators = LCM of 44, 20 and 10
2 44,20,10
2 22,10, 5
5 11,5,5
11, 1,1
LCM of denominators = 2x2x5x11 = 220
HCF of the fraction=4/220=1/55
