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

JEE Main · 2026 · 21 Jan 2026 · Shift 1 · Permutations and Combinations

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

Answer: 1333

Modulo

5

6^{m} \equiv 1

and

9^{n} \equiv (- 1)^{n}

, so

6^{m} + 9^{n} \equiv 0

requires

n

odd; with

50

choices for

m

and

25

odd values of

n

p = 1250

. For

q

m + n

(which ranges from

2

100

) must be a square of a prime, i.e.

4, 9, 25, 49

, contributing

3 + 8 + 24 + 48 = 83

pairs. Hence

p + q = 1250 + 83 = 1333

JEE Main · 2026 · 21 Jan 2026 · Shift 1 · Integer answer