Now taking s = a/q in (10), and summation for all 1 [less than or equal to] a [less than or equal to] q - 1 and noting the definition and properties of complete residue system mod q (see ) we have
Since (h, q) = 1, if a pass through a complete residue system mod q, then ha also pass through a complete residue system mod q.
To prove Theorem 2, we note that, for any odd number q [greater than or equal to] 3, if a pass through a complete residue system mod q, then 2a also pass through a complete residue system mod q.
If we replace the complete system of incongruent residues modulo m in this theorem by a reduced residue system modulo m, the result is not true.
Suppose that [a.sub.1], [a.sub.2], ..., [a.sub.[phi](m)] is a reduced residue system modulo m, then k[a.sub.1] + b, k[a.sub.2] + b, ..., k[a.sub.[phi](m)] + b is also a reduced residue system modulo m if and only if one of the following two conditions hold:
The number of distinct integers in a reduced residue system (mod n) is important here.
If S is a complete residue system (mod n), then we say that the set S', consisting of all elements of S except 0 (mod n), is a complete system of nonzero residues (mod n).
It is important to note that the CPU doesn't "know" what residue system is intended; it is wired to produce a double-word bit pattern for arithmetic operations on pairs of one-word integer bit patterns.
In the LPR system this is interpreted as 12 + 13 = 9, while the LAR interpretation is (-4) + (-3) = -7; in either case, the residue of [2.sup.4] is appropriate for that particular residue system. Results are similar for modern 32-bit (or 16-bit) CPU's: integer arithmetic produces the bit patterns consistent with arithmetic modulo [2.sup.32] (or [2.sup.16]), with interpretation in a particular residue system left to the user.
Weed control problems associated with crop residue systems
Pesticide loss with no-till or other high crop residue systems
often was less than 10 percent of conventional tillage (12,26).