50^ root of a number x is 12 and 50^ root of another number y is 18 .

Question

${50}^{}$ root of a number $x$ is 12 and ${50}^{}$ root of another number $y$ is 18 . Then the remainder obtained on dividing $(x+y)$ by 25 is \_\_\_\_\_\_\_\_\_\_\_\_.

Answer: 23

Accepted Answer

Given ${x}^{\frac{1}{50}}=12\Rightarrow x={12}^{50}$

${y}^{\frac{1}{50}}=18\Rightarrow y={18}^{50}$

$12\equiv 13$ (Mod 25)

${12}^{2}\equiv 19$ (Mod 25)

${12}^{3}\equiv -3$ (Mod 25)

${12}^{9}\equiv -2$ (Mod 25)

${12}^{10}\equiv -1$ (Mod 25)

${12}^{50}\equiv -1$ (Mod 25) ..... (i)

Now

$18\equiv 7$ (Mod 25)

${18}^{2}\equiv -1$ (Mod 25)

${18}^{-50}\equiv -1$ (Mod 25) ..... (ii)

$	herefore $ ${12}^{50}+{18}^{50}\equiv -2$ (Mod 25)

$\equiv 23$ (Mod 25)

$	herefore $ Answer = 23

Solutions · 1