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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let $S = {(m,n) : m,n \in {1,2,3,\ldots,50}}$. 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 ______
Solution
$S = {1,2,3,\ldots,50}$
$p = (6^m + 9^n)$ is divisible by $5$
No. of ways
$6^m = (5\lambda + 1)^m = 5k + 1$
$9^n = (10 - 1)^n = 10\mu - 1$ if $n$ is odd
$\Rightarrow n$ must be odd
$10\mu + 1$ if $n$ is even
$\Rightarrow$ No. of ways $= 50 \times 25 = 1250$
$q \Rightarrow (m+n)$ is square of a prime
$m+n = 4, 9, 25, 49$
No. of ways: $3, 8, 24, 48$
$q = 3 + 8 + 24 + 48 = 83$
$p + q = 1250 + 83 = 1333$
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