lcm and gcd questions and answers with simple tricks.

GCD ( HCF )and LCM questions and answers :

GCD stands for greatest common divisors.

Or HCF stands for highest common factor.

LCM stand for Lowest Common Multiple.

• LCM it a small number that can be divided completely by the other given numbers.
• GCD it is the largest number which can divided the given numbers.
• GCD is always smaller or equal to a smaller number.
• LCM is always equal to or larger than a large number.
• LCM of fraction :
• LCM = LCM of numerators / GCD of denominators.
• GCD of fraction :
• GCD = GCD of numerators / LCM of denominators.
• LCM of prime numbers is theirs multiplication.
• GCD of prime numbers is always one.
• Relation between LCM and GCD in two numbers is :
• First number × second number = LCM × GCD

Lowest Common Multiple ( LCM ) :

• LCM it a small number that can be divided completely by the other given numbers.

Methods of finding LCM :

List Method :

Question: Find the LCM 

1) 5 , 8.            2) 4, 6.           3) 3, 7
1) 4, 6

Ans. :

Multiplication table of 15 : 15, 30, 45, 60, 75, 90,105, .......

Multiplication table of 18 : 18, 36, 54, 72,90,108, .....

Common multiples of 15 and 18 : 90,.....

The smallest of these numbers 90.

Therefore, LCM 40.

Divisibility Method :

3 | 15 | 18

5 | 05 | 06

6 | 01 | 06
   | 01 | 01

   

Multiplication of 3 × 5 × 6 = 90.

Therefore, LCM 90.

2) 4, 6

Ans.

List method :

Multiplication table of 4 : 4, 8, 12, 16, 20, 24, .....

Multiplication table of 6 : 6, 12, 18, 24, 30, 36, .....

Common multiples of 4 and 6 : 12, 24, ....

The smallest of these numbers 12.

Therefore, LCM 12.

Divisibility Method :

2|  4 |  6

2|  2 |  3  

3|  1 |  1

Multiplication of 2 × 2 × 3 = 12

Therefore, LCM 12.

3) 3, 7.

Ans. : LCM of prime numbers is theirs multiplication.

Therefore, 3 × 7 = 21.

Greatest common divisors ( GCD ) :

• GCD it is the largest number which can divided the given numbers.
• GCD is always smaller or equal to a smaller number.

Methods of finding GCD :

List method :

Example :

1) Find the GCD of 6 and 8.

Ans. :

Divisors of 6 : 1, 2, 3, 6.

Divisors of 8 : 1, 2, 4, 8.

Of the common divisors 1 and 2, 2 is the biggest.

Therefore, GCD of 6 and 8 is 2.

Divisibility method :

2 | 6 | 8

   | 3 | 4

Multiplication of Common divisor is 2.

Therefore, GCD of 6 and 8 is 2.

2) Find the GCD of 20 and 30.

Ans. :

Divisors of 20 : 1, 2, 4, 5, 10, 20.

Divisors of 10 : 1, 2, 5, 10.

Of the common divisors 1, 2, 5 and 10, 10 is the biggest.

Therefore, the GCD of 20 and 30 is 10.
Divisibility Method :
2 | 20 | 30
5 | 10 | 15
   | 02 | 03
Multiplication of  Common divisors 2 × 5 = 10.
Therefore, GCD of 20 and 30 is 10.
3) Find the GCD of 5 and 11.
Ans. :
GCD of prime numbers is always one.
Therefore GCD of 5 and 11 is One.

LCM of fraction :

LCM = LCM of numerator / GCD of denominator.

Example :

Q. Find the LCM

1) 2/3, 4/9.              2) 10/12, 5/6.
1) 2/3, 4/9.

Ans. :

LCM of numerator values : 2, 4.

By divisibility method :

2 | 2 | 4

2 | 1 | 2

   | 1 | 1

Multiplication of 2 × 2 = 4.

Therefore, LCM of 2 and 4 is  4.

GCD of denominator values : 3, 9.
By divisibility method :
3 | 3 | 9
   | 1 | 3
Multiplication of Common divisor is 3
Therefore, GCD of 3 and 9 is 3.
Now, LCM of 2/3 and 4/9 is 4/3.

2) 10/12, 5/6.
Ans. :
LCM of numerator values : 10, 5
By divisibility method :
2 | 10 | 5
5 | 05 | 5
   | 01 | 1
Multiplication of 2 × 5 = 10.
Therefore, LCM of 10 and 5 is 10.
GCD of denominator values : 12, 6
By divisibility method :
2 | 12 | 6
3 | 06 | 3
   | 02 | 1
Multiplication of Common divisor is 2 × 3 =6
Therefore, GCD of 12 and 6 is 6.
Now, LCM of 10/12 and 5/6 is 10/6.

GCD of fraction :

• GCD = GCD of numerators / LCM of denominators.

Example :
Que To Find the GCD
1) 2/3, 4/9.                 2) 9/10, 12/16
1)2/3, 4/9
Ans. :
GCD of numerator values : 2, 4
By divisibility method :
2 | 2 | 4
   | 1 | 2
Multiplication of Common divisor is 2.
Therefore, GCD of 2 and 4 is 2.
LCM of denominator values : 3, 9
By divisibility method :
3 | 3 | 9
3 | 1 | 3
   | 1 | 1
Multiplication of 3 × 3 is 9.
Therefore, LCM of 3 and 9 is 9.
Now, GCD of 2/3 and 4/9 is 2/9.
2) 9/10, 12/16

Ans. :
GCD of numerator values : 9, 12
By divisibility method :
3 | 9 | 12
   | 3 | 04
Multiplication of Common divisor is 3.
Therefore, GCD of 9 and 12 is 3.
LCM of denominator values : 10, 16
By divisibility method :
2 | 10 | 16
2 | 05 | 08
2 | 05 | 04
2 | 05 | 02
5 | 05 | 01
   | 01 | 01
Multiplication of 2 × 2 × 2 × 2 × 5 = 80.
Therefore, LCM of 10 and 16 is 80.
Now, GCD of 9/10 and 12/16 is 3/ 80

• Relation between LCM and GCD in two numbers is :
• First number × second number = LCM × GCD

 Example:
1) The LCM of 10 and 15 is 30, so what is the GCD?
Ans. :
First number × Second number = LCM × GCD
10 × 15 = 30 × GCD
( 10 × 15 ) / 30 = GCD
Therefore, GCD = 5.
2) The GCD of 15 and 18 is 3, so what is the LCM?
Ans. :
First number × Second number = LCM × GCD
15 × 18 = LCM × 3
( 15 × 18 ) / 3 = LCM
Therefore, LCM = 90.
3) The two numbers are 4x and 10x, their GCD is 12 and LCM is 120, so x=?
Ans. :
First number × Second number = LCM × GCD
4x × 10x = 12 × 120
40 ( x × x ) = 12 × 120
x × x = ( 12 × 120 ) / 40
x  × x = 36
x × x = 6 × 6
Therefore, x = 6.

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