The number of elements in the set n : 10 n 100 and 3 n - 3 is a multiple of 7 is __________ . [2023]

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Solution

Q.
The number of elements in the set { $n \in ℕ : 10 \leq n \leq 100$ and $3^{n} - 3$ is a multiple of 7} is __________ . [2023]

Ans.
(15)

$3^{n} - 3 = 7 k, k \in ℤ$

$\Rightarrow 3^{n} = 7 k + 3, k \in ℤ$ $...(i)$

$Now, 3 \equiv 3 mod 7$

$3^{3} \equiv -1 mod 7$

$3^{6} \equiv 1 mod 7$

$3^{7} \equiv 3 mod 7$

$3^{13} \equiv 3 mod 7$

$∴ n = 1, 7, 13, \dots will satisfy equation (i)$

This forms an A.P. with common difference 6.

Now, $N \in [10, 100]$

$∴ n = 13, 19, \dots$

Now, $13 + (n - 1) \cdot 6 \leq 100 \Rightarrow 13 + 6 n - 6 \leq 100$

$\Rightarrow 6 n \leq 100 - 7 \Rightarrow 6 n \leq 93 \Rightarrow n \leq 15.5 \Rightarrow n = 15$

So, there are 15 such numbers.

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