A^B mod C = ( (A mod C)^B ) mod C

can any one please explain and provide the proof of this property of modulo arithmetic.

A^B mod C = ( (A mod C)^B ) mod C

can any one please explain and provide the proof of this property of modulo arithmetic.

This follows directly from the multiplication property :

(A * B)\% m = ((A\%m)*(B\%m))\% m

Applying this property with B = A ,

(A^2)\%m = ((A\%m)*(A\%m))\%m = ((A\%m)^2)\%m

By induction for any general value of n,

(A^n)\%m = ((A\%m)^n)\%m

For Proving multiplication property, you can do induction on the addition property. (A+B)\%m = ((A\%m + B\%m))\%m