JEE 2023 — Mathematics PYQ

Solution:

Humein $5^{99}(\mathrm{mod}11)$ nikalna hai.

1. Power pattern:

   $5^1\equiv 5(\mathrm{mod}11)$

   $5^2\equiv 25\equiv 3(\mathrm{mod}11)$

   $5^5\equiv 5^2\times 5^2\times 5^1\equiv 3\times 3\times 5\equiv 45\equiv 1(\mathrm{mod}11)$

2. Divide power by 5:

   $99=5\times 19+4$

   $$5^{99}=(5^5{)}^{19}\times 5^4$$
   $$5^{99}\equiv (1{)}^{19}\times (5^2{)}^2(\mathrm{mod}11)$$
   $$5^{99}\equiv 1\times 3^2\equiv 9(\mathrm{mod}11)$$

Answer: Remainder 9 hai.