Let . If the number of elements in S such that is a multiple of 5 is p and the number of elements in S such that is a square of a prime number is , then is equal to _______. [2026]

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Solution

Q.
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 _______. [2026]

Ans.
(1333)

$S = {1, 2, 3, \dots, 50}$

$p = (6^{m} + 9^{n}) is divisible by 5$

$No. of ways$

$6^{m} = {(5 λ + 1)}^{m} = 5 k + 1$

$9^{n} = {(10 - 1)}^{n} =$ $10 μ - 1$ if $n$ is odd

$\Rightarrow n must be odd$

$10 μ + 1$ if $n$ is even

$\Rightarrow No. of ways = 50 \times 25 = 1250$

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

$m + n = 4$ $m + n = 9$ $m + n = 25$ $m + n = 49$

No. of $↓$ $↓$ $↓$ $↓$

ways 3 8 24 48

$q = 3 + 8 + 24 + 48 = 83$

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

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