NUMBERS - Impotant notes/Formulae

(1) Sample Q:
(i) Find the remainder when (2222...upto...1000digits) is divided by 3125?
(ii) Find the remainder when (2222...upto...1000digits) is divided by 40?

Solution (i): Rem (Any number/5n) = Rem(Last 5 digits of the number/5n)
315 = 5n (Multiple of 5)
Therefore Ans = Rem (22222/3125)=347
Solution (ii): Rem (Any number/2) = Rem(Last 2 digits of the number/2)
40=2n (Multiple of 2)
Therefore Ans = Rem (22/40)=22


(2) Sample Q: How many perfect squares are there in the first 50 whole numbers?
Solution: The trap here is the word ewhole numbers which includes '0' as well.
Please note that the number '0' is also a perfect square (Since 0 = 0*0).

(3) Sample Q (Successive division): Find the number "N" which when divided successively by a,b,c leaves remainder x,y,z.
Solution: The general form of such a number "N" --> N = (abc)k + [(zb+y)a + x]

(4) The values of Rem [(a/N), (a^2/N), (a^3/N), (a^4/N).....] (where a,N are positive integers), follow a specific repetative pattern.
Eg: Rem [(4/9), (4^2/9), (4^3/9), (4^4/9)....] = Rem [(4/9), (16/9), (64/9), (256/9)....] = 4,7,1,4,7,1,....