Numbers - Prime numbers and Applications
Prime Factor:
A prime number which exactly divides a number is the prime factor of that number.
Factors of
42 : 1,2,3,6,7,14,21,42
Prime
factor of 42 : 2,3,7
Prime Factorization:
Fundamental Theorem of
Arithmetic: It
states that every natural number greater than 1 can be
written as a unique product of prime numbers. This theorem is also called the unique
factorization theorem or the unique-prime-factorization theorem.
How to write a number as a product of prime factors :
If the given number is a prime, we
need not write the number as a product of prime factors. Because it has no any other
prime number as its factor.
If the number is not a prime number,
A. Find the smallest prime number that exactly
divides the given number.
B. Divide the given number by the prime to obtain a quotient.
C. Divide the quotient with the prime number and if the quotient is not divisible by the prime number which is taken, take next prime number which exactly divides it
D. Repeat the process with the quotient until the quotient is itself a prime number
E.The prime factorization is given by the product of the primes used in the division process and the final prime quotient.
B. Divide the given number by the prime to obtain a quotient.
C. Divide the quotient with the prime number and if the quotient is not divisible by the prime number which is taken, take next prime number which exactly divides it
D. Repeat the process with the quotient until the quotient is itself a prime number
E.The prime factorization is given by the product of the primes used in the division process and the final prime quotient.
Example : 180 = 2 x 2 x 3 x 3 x 5
The number of factors of a given number:
How to find the number of factors of a number
Write the given
number as a product of prime factors.
Consider the number
N=ap × bq × cr ×… where a, b and c are prime
numbers and p, q and r are positive integers.
Then the number of
factors of ‘N’ is
N= (p+1)(q+1)(r+1)…..
Factors of 18 :
1,2,3,6,9,18
Example: Find the number of factors of 18?
Solution : 18=21×32
Number of factors=
(1+1) (2+1)
= (2×3) =6
Sum of the factors of a given number:
Finding the sum of the factors of a number:
Write the given
number as the product of prime factors
Consider the number
of N= ap × bq × cr ×… where a, b and c are prime
numbers and p, q and r are positive integers.
Then sum of the
factors= $ (\frac{a^{p+1}\text{–}1}{a-1})\times
)(\frac{b^{q+1}\text{–}1}{b-1})\times(\frac{c^{r+1}\text{–}1}{c-1})
)$
Example : Find the sum of the factors of 120
Write 120 as
product of prime factors => 120=23
× 31 x 51
Sum of factors = $ (\frac{2^{3+1}\text{–}1}{2-1})\times
)(\frac{3^{1+1}\text{–}1}{3-1})\times(\frac{5^{1+1}\text{–}1}{5-1}
)$
= $ (\frac{2^{4}\text{–}1}{1})\times
)(\frac{3^{2}\text{–}1}{2})\times(\frac{5^{2}\text{–}1}{4})
)$
= ($\frac{15}{1}$
) ×($ \frac{8}{2} $
) x ($ \frac{24}{4}$
) =15 x 4 x 6 = 360
The number of ways to express a number as a product of 2 factors:
To find the number of ways a number can be expressed as the product of
2 factors:
Express given
number into a product of prime factors.
Consider N= ap
× bq × cr ×… where a, b, c are primes and p, q, r
are positive integers.
Number of factors =
(p+1) (q+1) (r+1)….
Product of two
factors = $\frac{Number\:of\:factors}{2}$
In how many ways you can express 24 as a product of two of its
factors?
24 = 23 x
31
Number of factors
of 24 = (3 + 1) (1 + 1)= 8
So, the number of ways
to express 24 as a product of two factors = $\frac{8}{2}$
=4
The number of ways to express a number as a product of two co-primes :
First, write down
the given number as the product of prime factors.
Say N= ap ×
bq × cr ×… where a, b, c are primes and p, q, r are positive
integers.
Then the number of ways
N can be written as the product of two relatively primes = 2k-1 where k
is the number of prime factors.
Example: In how many ways 180 can be written as the product of two
of its co-prime factors?
Solution: Write
180 as the product of prime factors
180 = 22
x 32 x 51 .
The total number of
unique prime factors are 3.
Hence required
number of ways = 23-1 = 4.
Number of
co-primes that are less than the given number:
To find the number
of relatively prime numbers that are less than the given number, express the
given number as the product of prime factors.
N= ap ×
bq × cr ×… where a, b and c are primes and p, q and r are positive
integers
Then the number of
co-primes less than N
= n (1- $\frac{1}{a}$ ) (1- $\frac{1}{b}$)(1-$\frac{1}{c})$
= n (1- $\frac{1}{a}$ ) (1- $\frac{1}{b}$)(1-$\frac{1}{c})$
Example: Find the number of co-primes to 180 that are less than
180?
Solution: Expressing
180 as product of prime factors => 180 = 22 x 32 x 51
So the number of co-primes
that is less than 180 and co-prime to 180
= 180 x (1 – $\frac{1}{2}$) (1- $\frac{1}{3}$) (1 -$\frac{1}{5}$) =48.
Product of Factors of a number :
How to find the product of all the factors of a number :
To find the product
of all divisors of the number N, write N as the product of prime factors.
N= ap ×
bq × cr ×… where a, b and c are primes and p, q and r are positive integers
Find the number of
factors of N n(f) = (p+1)(q+1)(r+1)
Then, the product of
all the factors = N$\frac{n(f)}{2}$
Example: Find the product of all the factors of 120?
Solution : Write
the 120 into product of prime factors
120 = 23
x 3 x 5
Number of factors
of 120 n(f)= (3+1)(1+1)(1+1) = 4 x 2 x 2 =16
Product of factors
of 120 = 120$\frac{16}{2}$ = 1208
Finding
the Unit digit :
How to find
the unit digit in a product or an expression
Unit digit rules:
In this, there are four rules to find out the
unit digit of a given number.
Rule (1): The digits which are equal to the unit digit
of power anything.
The digits in this rule are 0, 1, 5, 6
01=0 11=1 51=5 61=6
02=0 12=1 52=20 62=36
03=0 13=1 53=125 63=216
------ ------ ------- --------
------ ------ ------- --------
Rule (2): Odd power unit digits and even power unit
digits.
The digits in this rule are 4, 9.
è Odd
powers of 4, the unit digit is 4
41=4
43=64
……….
……….
è Even
powers of 4. The unit digit is 6
42=16
44=256
………..
è Odd
powers of 9. The unit digit is 9.
91=9
93=729
…………
è Even
powers of 9. The unit digit, i.e., 1
92=81
94=6561
………….
Rule (3): This rule is in cyclic form of length ‘4’. The
digits in this rule are 2, 3, 7, 8….
21=2 31=3 71=7
22=4 32=9 72=49
23=8 33=27 73=343
24=16 34=81 74=2401
25=32 35=343 75=16807
81=8
82=64
83=512
84=4096
85=32768
Find the last digit
of 2343
Solution: Here
power is 343.
So divide 343 by 4
we will get remainder 3, so unit digit is the same as that of 23 = 8.