UPSC NDA I 2024 Mathematics Paper

To determine the remainder when $2^{120}$ is divided by 7 , we can use Fermat's Little Theorem. Fermat's Little Theorem states that if $p$ is a prime number and $a$ is an integer not divisible by $p$, then:$a^{p-1} \equiv 1 \pmod{p}$In this problem, $a=2$ and $p=7$. According to Fermat's Little Theorem:$2^{7-1} \equiv 1 \pmod{7}$or$2^6 \equiv 1 \pmod{7}$This tells us that every sixth power of 2 is congruent to 1 modulo 7. Therefore, we can express $2^{120}$ as a multiple of 6 :$2^{120} = (2^6)^{20}$Since $2^6 \equiv 1 \pmod{7}$, we can write:$(2^6)^{20} \equiv 1^{20} \pmod{7}$This simplifies to:$1^{20} \equiv 1 \pmod{7}$Thus, the remainder when $2^{120}$ is divided by 7 is 1 .