2106/53
= (252)2 x 22/53
= 252 x 252 x 4 /53
= 1 x 1 x 4 = 4 (Ans)
Therefore, 3 is the required remainder.
WILSON’S THEOREM:-
EULER’S THEOREM:-
According to this theorem, R[ aØ(N)/N ] = 1 ;
Where (i) a &N are co-prime numbers.
(ii) Ø(N) is called Euler Totient Function i.e. the no. of positive integers less than N that are co-prime to N.
NOTE:- Fermat's Theorem is a special case of Euler's Theorem.
Shortcut Method to find Ø(N) :-
We can find out
the value of Ø(N) by following 2
steps.
Step-1:- Firstly, express N in terms of prime factors
may be in powers of prime in factorization.
Let N = p1 x
p2 x p3 x …. And So
on.
Step-2:- Then find the value of N{(p1 -1)/p1} {(p2
-1)/p2} {(p3 -1)/p3} …. And so on.
This
value will be equal to Ø(N).
e.g. Ø(14) = 6
Since,the +ve
integers co-prime to 14 are listed as 1,3,5,9,11&13.
Shortcut
to find Ø(14) ;
Step-1:- 14 can be factorized as 14=2 x 7
Step-2:- 14(1/2)(6/7) = 6
Ø(60) = 16
Step-1:- 60 can be factorized as 60 =22 x 3
x 5
Step-2:- 14(1/2)(2/3)(4/5) = 16
Q. What is the remainder when 78 is divided by 15 ?
Sol:- 15 = 3 x 5 [ 7&15 are coprime no.s ]
Ø(15) = 15 (2/3)(4/5) = 8
Hence, remainder is 1.
Q. What is the remainder when 21004 is divided by 25 ?
Sol:- 25 = 52 [ 2&25 are coprime no.s ]
Ø(25) = 25 x (4/5) = 20
(21004 ) /25
= (220)50 x 24 / 25
= 1 x 16 =16
Hence, remainder is 16.
Q. What is the remainder when 773 is divided by 90 ?
Sol:- 90 = 2 x 32 x 5
[ 2&25 are coprime no.s ]
Ø(90) = 90 x (1/2)(2/3)(4/5) = 24
773/90 = (724)3 x 7/90
= 13 x 7 =7
Hence,remainder is 7.
Q. What is the remainder when 9102 is divided by 125 ?
Sol:- 125 = 53
[ 9&125 are coprime no.s ]
Ø(125) = 125 x (4/5) = 100
9102/125
= 9100 X 92
/125
= 1 x 81 = 81
Hence,remainder is 81.
Q. What is the remainder when 13194 is divided by 360 ?
Sol:- 360 = 5x32x23
[ 13&360 are coprime no.s ]
Ø(360) = 360 x (1/2)(2/3)(4/5) = 96
13194/360
= (1396)2X 132 /360
= 12 x 169 = 169
Hence,remainder is 169.
If you have any doubt, please let me know.