jmc

Taking the given equation modulo 1399 gives $$35\cdot 40\equiv 1(\mathrm{mod}1399),$$so we know that 35 is the multiplicative inverse to 40. We want to use this to find the multiplicative inverse to $4\cdot 40=160$, so we want to try to "divide" 35 by 4.

The difficulty in dividing by 4 is that 35 is odd. We do know, though, that $$35\equiv 35+1399\equiv 1434(\mathrm{mod}1399)$$and this number is even! Let's go even further, though, to find a multiple of 4: $$35\equiv 35+3\cdot 1399\equiv 4232(\mathrm{mod}1399).$$Factoring 4 we get $$35\equiv 4\cdot 1058(\mathrm{mod}1399).$$Finally we multiply by 40: $$1\equiv 40\cdot 35\equiv 40\cdot 4\cdot 1058\equiv 160\cdot 1058(\mathrm{mod}1399).$$This argument is inelegant. Let's write it in a more clear order: $$\begin{array}{rl}1058\cdot 160& \equiv 1058\cdot (4\cdot 40)\\ & \equiv (1058\cdot 4)\cdot 40\\ & \equiv 35\cdot 40\\ & \equiv 1(\mathrm{mod}1399).\end{array}$$The multiplicative inverse to 160 modulo 1399 is $\boxed{1058}$.