7. why should the remainder be less than the divisor?

Brief Explanations:

The division algorithm states that for any two integers (a) (dividend) and (b) (divisor, (b\neq0)), there exist unique integers (q) (quotient) and (r) (remainder) such that (a = bq + r), where (0\leq r<|b|). If the remainder (r) were greater than or equal to the divisor (b), then we could increase the quotient (q) by at least (1) (e.g., if (r = b), then (a=b(q + 1)+0); if (r>b), say (r=b + k) where (k>0), then (a=b(q + 1)+k) with the new remainder (k) which is less than (r) and still non - negative). So, by the principle of the uniqueness of the quotient and remainder in the division algorithm, the remainder must be less than the divisor.

Answer:

The remainder must be less than the divisor because of the uniqueness of the quotient and remainder in the division algorithm. If the remainder were greater than or equal to the divisor, the quotient could be increased further to get a remainder within the required range ((0\leq r<|b|)).