aspg-24-2723-1791482518 20261008180158 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Journal of Neutrosophic and Fuzzy Systems JNFS 2771-6430 2771-6449 2024 8 1 Rashel Abu Hakmeh Faculty of Science, Mutah University, Jordan Murhaf Obaidi Mustansiriah University, Department of Mathematics, Iraq The main goal of this work is to study the solutions of linear congruencies in the standard and n-standard fuzzy number theoretical systems, where we present necessary and sufficient conditions for fuzzy congruence to be solvable in these systems. Also, we provide the conditions of fuzzy nilpotency and fuzzy invertibility modulo integers with many illustrated examples. On the other hand, we suggest an algorithm to generate fuzzy Pythagoras triples and fuzzy Pythagoras quadruples in the standard fuzzy number theoretical system. 2024 18 22 2723 https://creativecommons.org/licenses/by/4.0/ 10.54216/JNFS.080102 https://www.americaspg.com/journal/24/article/2723 https://www.americaspg.com/storage/81714176630.pdf