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One method is to use modulo-arithmetic.
An example: To find (11^7) mod 13 do the following:
11^2 = 121 is congruent to 4(mod 13)
11^7 = ((11^2)^3) * 11
=> (4^3)*11 mod 13
=> (4^2)*4*11 mod 13
=> 16*4*11 mod 13
=> 3*4*11 mod 13 Using: (a * b) mod c = ((a mod c )* (b mod c))mod c
=> 132 mod 13
=> 2 mod 13
i.e. 11^ 7 mod 13 is 2.
Using the same procedure, we get (5^97) mod 52 = 5
5^3 = 125 is congruent to 21 mod 52
(5^97) mod 52 => ((5^3)^32)*5 mod 52
=> (21^32)*5 mod 52
Using 21^2 = 441 is congruent to 25 mod 52,
=>( (21^2)^16 )* 5 mod 52
=> (25^16)* 5 mod 52
Using 25^2 = 625 is congruent to 1 mod 52,
=> (1^8)* 5 mod 52
=> 5 mod 52
i.e. the remainder when 5^97 is divided by 52 is 5.
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