526. A six-digit number ends with the digit 7. If this digit is moved to the beginning of the number, the number increases by 5 times. What is this number?
The problem can be solved in two ways.

1) We will solve it as a problem of restoring the record.

![](https://cdn.mathpix.com/cropped/2024_05_21_475fdfbb473422773b9eg-108.jpg?height=141&width=167&top_left_y=569&top_left_x=956)

where the five-digit numbers denoted by asterisks in the first multiplier and in the product are the same. We sequentially obtain:

![](https://cdn.mathpix.com/cropped/2024_05_21_475fdfbb473422773b9eg-108.jpg?height=141&width=1123&top_left_y=809&top_left_x=475)

The original number turned out to be 142857.

2) Let the desired number be $10 x+7$, where $x$ is the number of tens in the number. We get the equation

$$
(10 x+7) 5=7 \cdot 10^{5}+x
$$

From this, we find $x$, and then calculate $10 x+7$:

$$
\begin{gathered}
50 x+35=700000+x, \quad 49 x=699965 \\
x=14285, \quad 10 x+7=142857
\end{gathered}
$$

Answer: 142857.

The number 142857, which resulted in the answer, often appears when solving similar problems. From it, by moving the last digit to the first place, we sequentially find five such six-digit numbers:

$$
714285, \quad 571428, \quad 857142, \quad 285714, \quad 428571.
$$

The factorization of the number 142857 into prime factors is as follows:

$$
142857=3^{3} \cdot 11 \cdot 13 \cdot 37
$$

Therefore, this number is divisible by 27. But all the above six-digit numbers are also divisible by 27, and some are divisible by 81. The number 142857 is divisible by 11, 13, and 37. But all the other six-digit numbers are also divisible by 11, 13, and 37! (Check or find the statement of problem 256 from § 8.) In addition, if the number 142857 is multiplied by the numbers 2, $3,4,5$, and 6, then the following numbers are obtained:

$$
285714, \quad 428571, \quad 571428, \quad 714285, \quad 857142
$$

(check!) - the same five numbers, only recorded in a different order. Finally,

$$
142857 \cdot 7=999999
$$

Therefore, 142857 is an unusual number, worthy of attention. It will meet us again.