Proving Divisibility Rule(heuristic) for 7,11,13 based on the last three digits of the integer

Correcting a minor sign error for divisibility Criterion 3 proof in Titu Andreescu's Number Theory, Structures, Examples, Problems Chapter 6
1001=7x11x13 prime factorization relates to decimal expansion of the natural number to be checked for divisibility by 7,11,13. @PapaFlammy69 has used the test for divisibility by 7 many times to the delight of his adoring fans. Papa Mathematician knows best!