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Special divisibility cases
  •  div by 7
num. + unit digit * 7
BEST::=> difference between group of 3 digit alternate bundle is either 0 or multiple of 3.
  • div by 17
num. - unit digit * 5
  • div by 13
num + unit digit * 4
  • div by 31
num - unit digit * 3
  • div by 29
num + unit digit * 1
`

  • Sum of 1st odd numbers==n^2;==>n=(last no.+1)/2;
  • Sum of 1st even number ==n(n+1)==>n=last no./2;
  • number of matches in knock out tournament is (n-1)


 multiplication of a number with unit digit is called osculator
Osculators are either +ve or -ve

 +ve Osculator 
The number ending with 9,19,29,39
then its osculator is 1 are (0+1=1),(1+1=2),(2+1=3),(3+1=4) respectively and so on

 for the number ending with 3 we multiply with whole number by 3 so that last number becomes 9 and apply the same rule
-ve Osculator 
for the number ending with 1 or 7 we apply the -ve osculators 

 For 1 ending digit just drop the last digit and remaining number is osculator like 11,21,51,111,1231.. so on its osculators are 1,2,11,123.. respectively and so on 

 For 7 ending digit multiply number by 3 so that last number becomes 1 and after that apply same rule

  • Square root shortcut
sqrt(any number ending with 1) resulting number's last number is either 1 or 9
//if last digit of square is 6 then 2nd last digit is always odd
//if last digit of square is 5 then 2nd last digit is always 2.

//if last digit of square is 1 then 2nd last digit is always even.

//if last digit of square is 4 then 2nd last digit is always even
trick***==4*4=16 i.e table of 4 is always last digit is even

//the zero at the end of square is always even & nonzero part is should be unperfect square
//the number of digits after decimal in a square is always even eg; 1.4884
//if number is ending with 2,3,7 or 8 then it is not a perfect square

·        For HCF first adjust and then find the HCF
·        For LCM first find LCM and then adjust
·        In LCM if reminder is common then Ans = LCM + Common Reminder
·        In LCM if difference is common then Ans  = LCM - Common Diff.
·        For AP Sn= n/2 [2a+(n-1)d]...........n= (LT-FT)/d
·        For AP AVG = (FT+LT)/2
HCF * LCM = Product
       
number of matches in knockout tournament is (n-1)

A.B is max at center and minimum at end for square it exactly opposite
A.B.C = 15===>1.1.13

Two consecutive even numbers HCF is always 2;
Two consecutive odd numbers HCF is always 1;
Two consecutive numbers HCF is always 1;

Let N be the greatest number will dividing X,Y,Z number and leaving same remainder then perform (Y-X),(Z-X) or(X-Y) and perform HCF

to find number of zeros in number then divide that number by 5 until quotient become less than 5;

Finding maximum power of number which will completely divide the some X! number and if number is prime then use X!/(given prime number)===>continue until < prime number for composite number take larger one means if 6 then 2*3 then take 3;

MIXTURE

 pure milk of X lit. is replaced with water with Y lit with n times then
       ML/MX = ((X-Y)/X)^n;
  







·        AVG
AVG of n even numbers is = n + 1;
AVG of n odd numberrs is = n;
·        NV = OA +/-(n+/-1)(diff.)............{NV=new value, OA= Old avg, n=no. of quantities}
·        AM = (X+Y)/2 (sum of all numbers including 0 divided by number of elements) GM = SQRT(XY) HM = 2XY/(X+Y)
·        EXPENDITURE(E) = PRICE(P) * CONSUMPTION(C)
·        CLOCK
·        RELATIVE SPEED OF CLOCK = 11/12 ms/min
·        IF U SEEN * 12/11(COINCIDENCE,OPPOSITE, I.E. IF QUE. CONTAIN FIND ANGLE BET.---TO---)

·        THE ANGLE BETWEEN H AND M CLOCK IS GIVEN BY 30H-5.5M
ONE SINGLE SERIES MAY BE COMBINATION OF TWO SERIES


***LCM of two numbers is taken as ex. 15,10-----> take smaller number here 10
10*1=10 not divisible by 15
10*2=20 not divisible by 15
10*3=30 divisible by 15
********So LCM is 30*******


TIME AND WORK

P1H1D1/P2H2D2=W1/W2.......if work is given

P1H1D1=P2H2D2...................if work is not given

P==>no. of Persons
H==>no. of Hours
D==>no. of Days
W==>Work

There are 204 squares presents in chase board

 1 hectare = 10000m^2

 time and work are always inversely proportional
for days it is LCM/(sum of days)
for work it is (sum of days)/LCM

#TRUE DISCOUNT(TD)#

 Present Worth(P.W.) = (100 * Amount)/(100+R*T)

 Present Worth(p.w) = (100 * TD)/(R*T)

 Run Rate(RR) = total run/total over



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