Let 'x' and 'y' represent the numbers.

The difference of x and y is 1 means:

y - x = 1

The sum is 17 means:

x + y = 17

We have the following system of equations:

y - x = 1

x + y = 17

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click here to see the system of equations solved for x and y

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x = 8

y = 9

The two numbers are 8 and 9.