JEE Main 2026 — Jan 21, Shift 1 — Mathematics solution

JEE Main2026Jan 21, Shift 1Mathematics

Let $S = {(m, n) : m, n \in {1, 2, 3, \dots, 50}}$ . If the number of elements $(m, n)$ in S such that $6^{m} + 9^{n}$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to……..

Your answer

Solution

Counting p: $6 \equiv 1 (mod 5)$ , so $6^{m} \equiv 1$ for every $m$ . $9 \equiv - 1 (mod 5)$ , so $9^{n} \equiv (- 1)^{n}$ . The sum is $\equiv 0 (mod 5)$ iff $n$ is odd. There are 50 choices for $m$ and 25 odd values of $n$ , giving $p = 50 \times 25 = 1250$ .

Counting q: $m + n$ ranges from 2 to 100, so squares of primes in this range are $4, 9, 25, 49$ . For sum $= s$ with $1 \leq m, n \leq 50$ , the count is $s - 1$ if $s \leq 51$ and $101 - s$ otherwise. So $q = 3 + 8 + 24 + 48 = 83$ .

$p + q = 1250 + 83 = 1333$ .

← Back to full paper

Practice more Mathematics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →