PRIME NUMBERS

            Here, you are going to study the following:
> what a prime number is.
> how to factorize a prime number
> calculating highest common factor(HCF or GCD) using prime factorization
> calculating lowest common multiples(LCM) using prime factorization

1) PRIME NUMBERS. Consider the numbers 1,2,3,4,5,6. the factors of these numbers can be written as follows:
D(1) ={1}
D(2)={1,2}
D(3)={1,3}
D(4)={1,2,4,}
D(5)={1,5}
D(6)={1,2,3,6}
   Note that 2,3 and 5 each has two factors while 4 and 6 have more than two factors while 1 has only one factor.The numbers with two factors are called Prime Numbers.

DEFINITION: A prime number is a number which has exactly two factors.For example {2,3,5,7,9,11,13,17,19,...}
    Numbers such as 4 and 6 above which have more than two factors are called Composite Numbers. 1 is neither a prime number nor a composite number because it has only one factor. 2 is the only prime number that is even.All other prime numbers are odd.

HOW TO FACTORIZE A PRIME NUMBER.We can factorize a prime number by using its prime factors. This can be done by listing out all the factors and then selecting those that are prime numbers OR by writing it as a product of its prime factors.Remember that the prime factors denoted by P of a number are the factors of that number that are prime numbers.
For example; 2 is a prime factor of 8. 4 is not a prime factor of 8.3 is a prime factor 12. 6 is not a prime factor of 12. You should be able to select all the prime factors of any given number.
     Example 1.List all the i) factors
                                      ii) prime factors of each of the following numbers.
      a) 12, b) 20
                               Solution
a) factors of 12; D(12)={1,2,3,4,6,12}
           prime factors of 12;P(12)={2,3}
      b) factors of 20; D(20)={1,2,4,5,10,20}
          prime factors of 20 P(20)={2,5}
Example 2. Factorize each of the following by writing as a product of its prime factors.
a) 4, b) 6, c) 8, d) 9,e) 18
                               Solution
a) 4=2× 2
b) 6=2× 3
c) 9=3×3
d) 18=2× 3×3
   Prime Factorization is finding the factors of a composite number from their prime products.Prime numbers cannot be factorized like composite numbers ; that is using products of prime numbers because they are themselves prime.
We can factorize bigger composite numbers by dividing continuously by successive prime factors.
Example 3. Factorize completely a) 1960 , b) 3410.
                  Solution
a)      2    1960
   2    980
   2    490
   5    245
   7    49
   7    7
   1
   1960= 2×2× 2×5×7×7

b) 2 3410
5    1705
   11 341
   31         31
   1
3410= 2× 5× 11× 31

        CALCULATING HCF or LCM OF TWO OR MORE NUMBERS BY PRIME FACTORIZATION
i) HIGHEST COMMON FACTOR( HCF)
To find the HCF or GCD of two numbers, do the following:
    > write down the two numbers
    >get prime factors common for each of the two numbers
    >write them as products of primes
    >Select the prime factors common in the numbers with the SMALLER (or SMALLEST) INDEX.
Example 4. Find the HCF of 1960 and 3410.
                          Solution
    1960= 2×2×2×5×7×7
    3410=2×5×11×31
   The prime numbers common to both products with the smaller indices are 2 and 5.
HCF( 1960, 3410)= 2×5
HCF(1960, 3410)=10. answer
Example 5. Find the GCD of 2×3×5×5×5 and 2×3×3×5×5×7×7
   GCD= 2×3×5×5
   GCD=150. answer
Example 6: Find the HCF of 168 and 180.
                        Solution
   Get prime factors common to the two numbers that divide them exactly.Stop where a prime number cannot divide simultaneously the numbers under consideration.
         2         168        180
         2           84         90
         3    42 45
                      14         15
HCF( 168,180)= 2×2×3
HCF(168,180)=12. answer

ii) LOWEST COMMON MULTIPLE( LCM)
To find the LCM of two or more numbers, do the following:
   >get the prime factors common to the numbers
   >Divide with successive prime numbers right down to 1 for all the numbers being considered.
   >Select the first prime factors in common in the numbers with a LARGER (or LARGEST) index.
   Example 1: Find the LCM of 168 and 180.
Solution
   2 168 180
   2    84 90
     2              42          45
     3             21           45
     3              7             15
     5               7             5
     7 7    1
     1           1
LCM(168, 180)= 2×2×2×3×3×3×5×7
   Example 2: Find the LCM of 3×3×5×5×5 and 2×3×3×5×7×7
   Solution
Select the first prime factors in common in the numbers with a LARGER (or LARGEST) index.
   LCM =2×3×3×5×5×5×7×7
Exercise.
1) Find the set of prime factors of each of the following
a) 12, b) 30, c) 45, d) 60
2) Find the HCF of the following:
a) (6,8) , b) (6,8,12) c) (9,12,30)
3) Find the LCM of (a), (b), and (c) above.
4) By leaving your answer in the index form , find the following:
   a) HCF(245,385) , b) HCF (360,420), c) LCM(245,385), d)LCM(360,420)
   e) LCM(8,10,12,18)
5) Find, a) HCF of 2×5×7×7, 2×2×3×5 and 2×3×3×7
              b)LCM of 2×2×2×2×3×3×7, 3×3×5×5×7 and 2×2×2×3×5×5
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