7-digit numbers are formed by the digits 1, 2, 3, 4, 5, 6, 7. In each number, no digit is repeated. Prove that among all these numbers, there is no number, which is a multiple of another number.
Total no. of ways of arranging it is 7!= 5040. So now we cannot check cases.
Suppose $x$ and $y$ are 2 nos. formed from the given digits with $x|y$ Then $y= 6x,5x,4x,3x,2x$.
But this leads nowhere...