What will be the remainder if the number $7 ^ { 2015 }$ is divided by $25$ ?

To find the remainder of 7^2015 divided by 25, we use Euler's theorem. Since the greatest common divisor of 7 and 25 is 1, 7 raised to the Euler totient function phi(25) leaves a remainder of 1 when divided by 25. Calculating phi(25) gives 25 multiplied by (4/5), which equals 20, so 7^20 is congruent to 1 modulo 25. Dividing the exponent 2015 by 20 gives a remainder of 15, meaning 7^2015 has the same remainder as 7^15. Computing 7^4 gives 2401, which has a remainder of 1, so 7^12 also has a remainder of 1; multiplying this by the remaining 7^3, which is 343, yields a final remainder of 18.

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