If the exact division algorithm is done with a full subtraction at each stage
and the dividend isn’t a multiple of the divisor, then low zero limbs are
produced but with a remainder in the high limbs. For dividend *a*,
divisor *d*, quotient *q*, and *b = 2^mp_bits_per_limb*, this remainder
*r* is of the form

a = q*d + r*b^n

*n* represents the number of zero limbs produced by the subtractions,
that being the number of limbs produced for *q*. *r* will be in the
range *0<=r<d* and can be viewed as a remainder, but one shifted up by
a factor of *b^n*.

Carrying out full subtractions at each stage means the same number of cross
products must be done as a normal division, but there’s still some single limb
divisions saved. When *d* is a single limb some simplifications arise,
providing good speedups on a number of processors.

The functions `mpn_divexact_by3`

, `mpn_modexact_1_odd`

and the
internal `mpn_redc_X`

functions differ subtly in how they return *r*,
leading to some negations in the above formula, but all are essentially the
same.

Clearly *r* is zero when *a* is a multiple of *d*, and this
leads to divisibility or congruence tests which are potentially more efficient
than a normal division.

The factor of *b^n* on *r* can be ignored in a GCD when *d* is
odd, hence the use of `mpn_modexact_1_odd`

by `mpn_gcd_1`

and
`mpz_kronecker_ui`

etc (see Greatest Common Divisor).

Montgomery’s REDC method for modular multiplications uses operands of the form
of *x*b^-n* and *y*b^-n* and on calculating *(x*b^-n)*(y*b^-n)* uses the factor of *b^n* in the exact
remainder to reach a product in the same form *(x*y)*b^-n*
(see Modular Powering).

Notice that *r* generally gives no useful information about the ordinary
remainder *a mod d* since *b^n mod d* could be anything. If
however *b^n ≡ 1 mod d*, then *r* is the negative of the
ordinary remainder. This occurs whenever *d* is a factor of
*b^n-1*, as for example with 3 in `mpn_divexact_by3`

. For a 32 or
64 bit limb other such factors include 5, 17 and 257, but no particular use
has been found for this.