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[The Problem of the Week] Square squared and more

Another published in Dodka old problems, which, as usual, I lost the source and I can not say now where you extracted.

I took a number and found its square. After I have raised this square to square and multiply the result by the original number. At the end of my calculations, I find as a result a number of 7 digits finish 7. What is the original number?

The solution will not be expected, as the image let it pass.

[This image is taken from the web Grand Illusions. Consists of a simple toy consisting of six squares made of clear plastic with three colors, and joined by a corner with a piece of rubber that can be pasted to a window, for example. With the light coming through the window, the children can play to turn over to the squares and discover the different shades of colors that are formed by combining the squares together. I recommend visit grand-illusions.com Because it is full of all sorts of objects and toys magic, optical illusions, and articles and videos of many of them]

Solution:

If we square, then back to square and then multiply the original number, we are bringing to the fifth:

(x ² ) ² · x = x ⁵

We have to find a number raised to the fifth of a result seven digits ending in 7. This can be done to score, but it is sufficient to prove with numbers ending in 7, because only they can to last digit 7 as they rise to the fifth power (those ending in 3 also can give 7 in the last figure, but not when they rise to the fifth power).

After some trial and error is found that 17 ⁵ = 1419857, ie 17 is the only number it does.