What is the remainder of the following? $$\dfrac{55^{75}×93^{175}×107^{275}}{17}$$
My solution:
The remainder of $\dfrac{55^{75}}{17}$ is $13$.
The remainder of $\dfrac{93^{175}}{17}$ is $15$.
The remainder of $\dfrac{107^{275}}{17}$ is $6$.
Finally, the remainder of $\dfrac{13×15×6}{17}$ is $14$, which must be the solution.
Please suggest whether this is correct.
Say that $a$ leaves a remainder of $x$ and $b$ leaves a remainder of $y.$ So, we can write
$$a = 17n + x,$$ $$b = 17m + y.$$
Thus,
$$ab = (17n+x)(17m+y) = 17^2nm + 17(ny + mx) + xy,$$
so $ab$ leaves the same remainder as $xy$ when you divide it by $17.$
Luckily for you, you have a product and you've computed the remainder of each individual term--so you can just multiply them, as you did, to get the remainder that the entire product leaves.