Let S=\ (m, n): m, n in\ 1,2,3, ldots ldots, 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... \ \ \ \

Mathematics · Permutations and Combinations

JEE Main 2026 — 21 January, Morning Shift — Question 23

Let $S=\{(m, n): m, n \in\{1,2,3, \ldots \ldots, 50\}\}$ . 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... $\_\_\_\_$

Answer: 1333

Numerical answer — enter this value.

Step-by-step solution

$\mathrm{S}=\{1,2,3, \ldots .50\}$

$\mathrm{p}=\left(6^{\mathrm{m}}+9^{\mathrm{n}}\right)$ is divisible by 5 No. of ways

$6^{\mathrm{m}}=(5 \lambda+1)^{\mathrm{m}}=5 \mathrm{k}+1$

$9^{n}=(10-1)^{n}=10 \mu-1$ if $n$ is odd ⇒ n must be odd

$10 \mu+1$ if n is even ⇒ No. of ways $=50 \times 25=1250$

$\mathrm{q} \Rightarrow(\mathrm{m}+\mathrm{n})$ is square of a prime

	$m+n=4$	$m+n=9$	$m+n=25$	$m+n=49$
No. of	$\downarrow$	$\downarrow$	$\downarrow$	$\downarrow$
Ways	3	8	24	48

q = 3 + 8 + 24 + 48 = 83

p + q = 1250 + 83 = 1333

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 21 January, Morning Shift
Subject: Mathematics
Chapter: Permutations and Combinations
Topic: Applications of Permuations and Combination

The number of strictly increasing functions f from the set \1,2,3,4,5,6\ to the set (1,2,3, ldots, 9) such that f( i) neq i for 1 leq i leq…

Answer: 1333

Step-by-step solution

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