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

Let S m n m n in 1 2 3 50 If the number of elements m n... | null (Mathematics)

JEE Mains 2026 21st January Shift 1

Mathematics

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

\_\_\_\_

\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^{\mathrm{n}}=(10-1)^{\mathrm{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

\\ TABLE

\\ \mathrm{q}=3+8+24+48=83\\

=\mathrm{p}+\mathrm{q}=1250+83=1333

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