I recently came across this test here
I understand how this works but I noticed something in the example given in said answer for the mod 7 of 43211 (I would assume this pattern extends to all other numbers).
(when 43211 is being simplified to 1211) 43000 ≡ 1000 (mod 700)
Subsequently,
(when 1211 is being simplified to 511) 1200 ≡ 500 (mod 70)
Again,
(when 511 is being simplified to 21) 510 ≡ 20 (mod 7)
I can see why this pattern would arise considering how the method was proved in the first place (using radix [being 10] polynomials in the nested Horner form) but a more mathematical reason or intuition would be greatly appreciated