Other Approach.?

Hello friends, I solved this question but I want to know other method which u think will be the shortest method. Find the number of positive integral values of $x \leq 100$ such that $3^{x} - x^{2}$ is divisible by 5.I think answer is 20.

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

We are looking for the values of $x$ such that $3^x-x^2=5k$ with $k\in\mathbb{Z}$ , or $3^x \equiv x^2 \pmod{5}$ . The sequence $a_x = 3^x \pmod{5}$ for $1 \leq x \leq 100$ is $3,4,2,1,3,4,2,1,\dots$ with a period of 4. The sequence $b_x = x^2 \pmod{5}$ for $1 \leq x \leq 100$ is $1,4,4,1,0,1,4,4,1,0,\dots$ with a period of 5. So, if $3^x \equiv x^2 \pmod{5} \implies a_x=b_x$ , then either $a_x=b_x=1$ or $a_x=b_x=4$ .

Case 1: $a_x=b_x=1$ . $a_x = 1$ whenever $x \equiv 0 \pmod{4}$ and $b_x = 1$ whenever $x \equiv 1 \pmod{5}$ or $x \equiv 4 \pmod{5}$ .

$\bullet$ Case 1a) $x \equiv 0 \pmod{4}$ and $x \equiv 1 \pmod{5} \implies x \equiv 16 \pmod{20} \implies x \in \{16,36,56,76,96\}$ .

$\bullet$ Case 1b) $x \equiv 0 \pmod{4}$ and $x \equiv 4 \pmod{5} \implies x \equiv 4 \pmod{20} \implies x \in \{4,24,44,64,84\}$ .

Case 2: $a_x=b_x=4$ . $a_x = 4$ whenever $x \equiv 2 \pmod{4}$ and $b_x = 4$ whenever $x \equiv 2 \pmod{5}$ or $x \equiv 3 \pmod{5}$ .

$\bullet$ Case 2a) $x \equiv 2 \pmod{4}$ and $x \equiv 2 \pmod{5} \implies x \equiv 2 \pmod{20} \implies x \in \{2,22,42,62,82\}$ .

$\bullet$ Case 2b) $x \equiv 2 \pmod{4}$ and $x \equiv 3 \pmod{5} \implies x \equiv 18 \pmod{20} \implies x \in \{18,38,58,78,98\}$ .

So, $x \in \{ 2,4,16,18,22,24,36,38,42,44,56,58,62,64,76,78,82,84,96,98 \}$ , which are indeed 20 values.

Tim Vermeulen - 7 years, 11 months ago

I did the same but I want any other short method.Thanks for ur efforts......

Kiran Patel - 7 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$