`(quotient n[1] n[2])`procedure`(remainder n[1] n[2])`procedure`(modulo n[1] n[2])`procedureThese procedures implement number-theoretic (integer) division. n[2] should be non-zero. All three procedures return integers. If n[1]/n[2] is an integer:

(quotient n[1] n[2]) ===> n[1]/n[2] (remainder n[1] n[2]) ===> 0 (modulo n[1] n[2]) ===> 0

If n[1]/n[2] is not an integer:

(quotient n[1] n[2]) ===> n[q] (remainder n[1] n[2]) ===> n[r] (modulo n[1] n[2]) ===> n[m]

where n[q] is n[1]/n[2] rounded towards zero, 0 < |n[r]| < |n[2]|, 0 < |n[m]| < |n[2]|, n[r] and n[m] differ from n[1] by a multiple of n[2], n[r] has the same sign as n[1], and n[m] has the same sign as n[2].

From this we can conclude that for integers n[1] and n[2] with n[2] not equal to 0,

(= n[1] (+ (* n[2] (quotient n[1] n[2])) (remainder n[1] n[2]))) ===> #t

provided all numbers involved in that computation are exact.

(modulo 13 4) ===> 1 (remainder 13 4) ===> 1 (modulo -13 4) ===> 3 (remainder -13 4) ===> -1 (modulo 13 -4) ===> -3 (remainder 13 -4) ===> 1 (modulo -13 -4) ===> -1 (remainder -13 -4) ===> -1 (remainder -13 -4.0) ===> -1.0 ; inexact