Da Form 15 6: Fill & Download for Free

You don’t really even need a scientific calculator for something like this. You only need some knowledge of how fast modular exponentiation works. Here is a fairly simple algorithm.First, begin by splitting apart into square powers, until we find something very simple that we can calculate easily. Here, this works as follows.[math]\begin{align*} 7^{15} &= 7 \cdot \left(7^7\right)^2 [/math][math][/math][math]\\ 7^7 &= 7 \cdot \left(7^3\right)^2 [/math][math][/math][math]\\ 7^3 &= 7 \cdot 7^2 \end{align*}.\tag*{}[/math]We can now run this in reverse, doing very easy calculations at each step.[math]\begin{align*} 7^2 &= 49 = 3 \mod 23 [/math][math][/math][math]\\ 7^3 &= 7 \cdot 3 = -2 \mod 23 [/math][math][/math][math]\\ 7^6 &= (-2)^2 = 4 \mod 23 [/math][math][/math][math]\\ 7^7 &= 7 \cdot 4 = 5 \mod 23 [/math][math][/math][math]\\ 7^{14} &= 5^2 = 2 \mod 23 [/math][math][/math][math]\\ 7^{15} &= 7 \cdot 2 = 14 \mod 23 \end{align*}.\tag*{}[/math]Ta da! There is arguably an even faster way if you can recognize that [math]7\cdot 10 = 1 \mod 23[/math]. Knowing that, it follows that [math]7^{15} = 7^{16} \cdot 10[/math]. However, since [math]16[/math] is a power of [math]2[/math], it is significantly easier to calculate via squaring.[math]\begin{align*} 7^{16} &= \left(7^8\right)^2 [/math][math][/math][math]\\ 7^8 &= \left(7^4\right)^2 [/math][math][/math][math]\\ 7^4 &= \left(7^2\right)^2 \end{align*} \tag*{}.[/math]We already know that [math]7^2 = 3 \mod 23[/math]. Consequently,[math]\begin{align*} 7^4 &= 3^2 = 9 \mod 23 [/math][math][/math][math]\\ 7^8 &= 9^2 = -11 \mod 23 [/math][math][/math][math]\\ 7^{16} &= \left(-11\right)^2 = [/math][math]6[/math][math] \mod 23 \end{align*}.\tag*{}[/math]Therefore,[math]\begin{align*} 7^{15} &= [/math][math]6[/math][math] \cdot 10 = 14 \mod 23 \end{align*}.\tag*{}[/math]The point is that by splitting apart into squares, you only ever do fairly non-intensive calculations, and you don’t have an overwhelming number of them, since you only have as many steps as the number of times it takes to divide the initial exponent by [math]2[/math] before it gets to [math]1[/math]. This basic idea is one of the components that allows cryptosystems like RSA to run in a reasonable amount of time.

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