The number of elements in the set {n N : 10 n 100 and 3? - 3 is a multiple of 7} is

Question

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 __________.

Smart Achievers · Accepted Answer

(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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