Confusion about Modulus Multiplication and Exponents

Solution 1:

I can confirm, as hinted in a comment to your question, that the discrepancy arises from a misinterpretation of the notation $$ 1847 (1002, 493) \bmod 2503. \tag{multiple} $$

Wolfram understands this as $$ (1847 * 1002 \bmod 2503, 1847 * 493 \bmod 2503). $$ Fair enough - how could Wolfram know about the elliptic curve you have in mind?

The correct interpretation in the exercise is that (multiple) refers to the multiple with respect to addition on the elliptic curve. Please check the formulas for point addition (and doubling) on an elliptic curve.

If you wish I can provide simple GAP code for point addition and doubling.