Common Factor

Common Factor

A common factor of two or more numbers is a number which divides each of them exactly.
For example, 4 is a common factor of 8 and 12.

Highest common factor
Highest common factor of two or more numbers is the greatest number that divides each one of them exactly. For example, 6 is the highest common factor of 12, 18 and 24. Highest Common Factor is also called Greatest Common Divisor or Greatest Common Measure.
Symbolically, these can be written as H.C.F. or G.C.D. or G.C.M., respectively

Methods of Finding H.C.F.

 I. Method of Prime Factors
Step 1 Express each one of the given numbers as the product of prime factors.
[A number is said to be a prime number if it is exactly divisible by 1 and itself but not by any other number, e.g. 2, 3,
5, 7, etc. are prime numbers]
Step 2 Choose Common Factors.

Step 3 Find the product of lowest powers of the common factors. This is the required H.C.F. of given numbers.

 Illustration 1 Find the H.C.F. of 70 and 90.
Solution 70 = 2 x5 x 7
90 = 2 x 5 x 9
Common factors are 2 and 5.

 H.C.F. = 2 x 5 = 10.

 Illustration 2 Find the H.C.F. of 3332, 3724 and 4508
Solution 3332 = 2x 2x 7x 7 x 17
3724 = 2 x 2 x 7 x 7 x 19
4508 = 2 x 2 x 7 x7 x 23

 H.C.F. = 2 x 2 x 7 x 7 = 196.

 Illustration 3 Find the H.C.F. of 360 and 132.
Solution 360 = 2x2x2x3x3x 5
132 = 2x2 x 3x1 x 11

 H.C.F. = 2x2 x 3x1  = 12.

 II. Method of Division
A. For two numbers:
Step 1 Greater number is divided by the smaller one.
Step 2 Divisor of (1) is divided by its remainder.
Step 3 Divisor of (2) is divided by its remainder. This is continued until no remainder is left.

H.C.F. is the divisor of last step.

                         Find the H.C.F. of 3556 and 3444
                                3444 )3556 (1
                                          3444
                                            112 ) 3444 ( 30
                                                     3360
                                                84 ) 112 ( 1
                                                          84
                                                   28 ) 84 ( 3
                                                          84
´                                                          0

 B. For more than two numbers:
Step 1 Any two numbers are chosen and their H.C.F. is obtained.
Step 2 H.C.F. of H.C.F. (of(1)) and any other number is obtained.
Step 3 H.C.F. of H.C.F. (of (2)) and any other number (not chosen earlier) is obtained.
This process is continued until all numbers have been chosen. H.C.F. of last step is the required H.C.F.
Illustration 6 Find the greatest possible length which can be used to measure exactly the lengths
7 m, 3 m 85 cm, 12 m 95 cm.
Solution Required length

= (H.C.F. of 700, 385, 1295) cm = 35 cm.

 Common Multiple
A common multiple of two or more numbers is a number which is exactly divisible by each one of them.

For Example, 32 is a common multiple of 8 and 16.

 8 x 4 = 32

16 x 2 = 32.

Least Common Multiple

The least common multiple of two or more given numbers is the least or lowest number which is exactly divisible by
each of them.
For example, consider the two numbers 12 and 18.
Multiples of 12 are 12, 24, 36, 48, 72, …
Multiple of 18 are 18, 36, 54, 72, …
Common multiples are 36, 72, …

\Least common multiple, i.e. L.C.M. of 12 and 18 is 36.
Methods of Finding L.C.M.

 A. Method of Prime Factors
Step 1 Resolve each given number into prime factors.
Step 2 Take out all factors with highest powers that occur in given numbers.

Step 3 Find the product of these factors. This product will be the L.C.M.

 Find the L.C.M. of 32, 48, 60 and 320.
Solution 32 = 25 x 1
48 = 24 x 3
60 = 22 x 3 x 5
320 = 26 x 6
\ L.C.M. = 26 x 3 x 5 = 960.

 B. Method of Division
Step 1 The given numbers are written in a line separated by common.
Step 2 Divide by any one of the prime numbers 2, 3, 5, 7, 11, … which will divide at least any two of the given
numbers exactly. The quotients and the undivided numbers are written in a line below the first.
Step 3 Step 2 is repeated until a line of numbers (prime to each other) appears.

1 Find the product of all divisors and numbers in the last line which is the required L.C.M.

 Find the L.C.M. of 12, 15, 20 and 54.
Solution 2   12 , 15, 20 , 54
               2    6 , 15 , 10 , 27
              3     3 , 15 , 5 , 27           
              5     1 , 5 , 5 , 9

                     1,  1, 1, 9  
L.C.M. = 2 ´ 2 ´ 3 ´ 5 ´ 1 ´ 1 ´ 1 ´ 9 = 540.
Note: Before finding the L.C.M. or H.C.F., we must ensure that all quantities are expressed in the same unit.              

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