NUMBERS:An important QUAT topic
SQUARES
1. Square of a number ending with 5
Let y5 be the number then
square is :- y*(y+1)25
Eg :-
35 * 35 = 1225
3*(3+1)25=1225
2. Square of a number near to a known square
This is real simple you just need to look around ,the formula is very familiar.
(n+1)^2 = n^2 +2*n +1
Eg:-
35^2 can be found easily,so 34^2 = 1225-2*34-1=1156
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To find the digit in the unit place of a number given in powers???
Have a look at the pattern below,here the case is with “powers of 2″
2^1 = 2
2^2 =4
2^3 =8
2^4=16
2^5=32
2^6=64
If you have carefully look at the above table you can find that the digits in the unit place forms a pattern(2,4,8,6 ) which is repeating.From this we can deduce that 2^8 will end with unit place as 6.Similarly 2^10 will end with unit place as 4.
Similarly for every numbers by checking the pattern we can easily find the unit place digit.
Eg:-What will be the unit place digit of 3^46.
3^1=3
3^2=9
3^3=27
3^4=81
3^5=243
3,9,7,1 is the reapeating pattern.
So 3^46 will end with “9″.If you didn’t understand then read more.At multiples of “4″ the power of three will give 1 as unit place digit.
So 3^44 will give “one” at unit place.Therefore 3^46 will give “9″.
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Sum of all factors of a number “N”
Represent the number “N” as a product of prime numbers.
(a^p) *( b^q)*(c^r)
Formula for sum is given below.
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What is a CO-PRIME??
Two numbers are said to be Co-Primes if they DONT have any common factors other than one.
Number of ways 2 numbers can be written as product of Co-Primes
is 2^(n-1),where n is a distinct prime number.
Eg:-Question:-How many ways 48 can be written as a product Co-Primes??
Solution:- Write 48 as a product of prime powers
2^4 * 3^1=16*3=48
We can write 48 as product of two prime powers,therefore n=2.
No: of ways=2^(2-1)=2
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Find number of Co-Primes to N that are less than N.
Represent N as (a^p)*(b^q)*(c^r),where a,b,c are prime numbers.
No of Co-Primes less than N = N*[1-1/a]*[1-1/b]*[1-1/c]
Eg: Question : find no: of Co-Primes till 165
165=3*5*11
Therefore answer = 165*{1-1/3}{1-1/5}{1-1/11}=80
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The sum of Co-Primes less than N =(N/2)*No of Co-Primes less than N
Eg:Sum of Co-Primes till 165??
(165/2)*80=6600
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