A number has 6 different digits. If the last digit is moved to the front, then a new number is formed which is exaclty 5 times the old number. With which number did we start?

Write down the digits in the first number as ABCDEF. The new number is hence to be written as FABCDE. We define G to be ABCDE.
So: 100000F + G = 5 (10G + F)
99995F = 49G
Division by 7 gives 14285F = 7G
Because F is a number with 1 digit, the solutions is G = 14285 and F = 7. We started with 142857.

Remark: This number has some other nice properties. Multiply it with 2, 3, 4, 6 or 7! Or: calculate 1/7 in decimal form.