Conditions for a Zero Sum Modulo n
Canadian mathematical bulletin, Tome 18 (1975) no. 1, pp. 27-29
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In this paper the following result is proved.Let n > 0 and k ≥ 0 be integers with n — 2k ≥ 1. Given any n — k integers there is a non-empty subset of indices I ⊂ {1, 2,..., n — k} such that the sum Σi∊I ≡ 0(mod n) if at most n — 2k of the integers (1) lie in the same residue class modulo n.The result is best possible if n ≥ 3k — 2 in the sense that if "at most n — 2k" is replaced by "at most n — 2k + 1" the result becomes false. This can be seen by taking aj = 1 for 1 ≤ j ≤ n — 2k + 1 and aj = 2 for n —2k + 2 ≤ j ≤ n — k, noting that the number of 2's here is n — k — (n — 2k + 1)= k — 1 ≤ n — 2k + 1.
Bovey, J. D.; Erdös, Paul; Niven, Ivan. Conditions for a Zero Sum Modulo n. Canadian mathematical bulletin, Tome 18 (1975) no. 1, pp. 27-29. doi: 10.4153/CMB-1975-004-4
@article{10_4153_CMB_1975_004_4,
author = {Bovey, J. D. and Erd\"os, Paul and Niven, Ivan},
title = {Conditions for a {Zero} {Sum} {Modulo} n},
journal = {Canadian mathematical bulletin},
pages = {27--29},
year = {1975},
volume = {18},
number = {1},
doi = {10.4153/CMB-1975-004-4},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1975-004-4/}
}
TY - JOUR AU - Bovey, J. D. AU - Erdös, Paul AU - Niven, Ivan TI - Conditions for a Zero Sum Modulo n JO - Canadian mathematical bulletin PY - 1975 SP - 27 EP - 29 VL - 18 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1975-004-4/ DO - 10.4153/CMB-1975-004-4 ID - 10_4153_CMB_1975_004_4 ER -
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