Suppose that the integer is 123. If you place the digit 4 in front of it, you get the number 4123. If you place the 4 at the end of it, you get 1234. But 4123 is not equal to 4 times 1234. So 123 is not the correct solution.

In fact, the smallest solution is 10256, and 410256 = 4×102564. To see how to get the solution, see the similar problem here.