A Novel Divisibility Test for 7 Based on the Transformation N ′ = 4a − b

A Novel Divisibility Test for 7 Based on the Transformation N ′ = 4a − b | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 13 October 2025 V1 Latest version Share on A Novel Divisibility Test for 7 Based on the Transformation N ′ = 4a − b Author : Abiud Wakhanu Mulongo 0009-0003-4435-9891 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176038359.95362796/v1 206 views 116 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Divisibility tests provide efficient, often mental, methods for determining whether integers are divisible by given primes. While many rules for divisibility by 7 exist, including subtraction, addition, modular arithmetic, and block methods, they are often computationally awkward or cognitively demanding. This paper introduces and formalizes a new divisibility test for 7 discovered independently by the author in 2000: for a number N = 10a + b, where a is the truncated integer without the last digit and b is the final digit, N is divisible by 7 if and only if N ′ = 4a − b is divisible by 7. The rule is shown to be mathematically equivalent to classical modular rules, yet it introduces a novel multiplier of 4 that has not appeared in traditional arithmetic textbooks. We provide proof of validity, illustrative examples, computational analysis, raw performance results across 3-6 digit numbers, and a comparative evaluation of efficiency and cognitive complexity. This contribution demonstrates that even within elementary number theory, new yet simple approaches can be found that enrich pedagogy, expand cognitive capabilities, and inform lightweight modular arithmetic in engineering. Supplementary Material File (abiud_m_wakhanu_dvisibility_test_for_number_7_25092025.pdf) Download 227.40 KB Information & Authors Information Version history V1 Version 1 13 October 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords cognitive development cryptography divisibility tests modular arithmetic number theory pedagogy seven signal processing Authors Affiliations Abiud Wakhanu Mulongo 0009-0003-4435-9891 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 206 views 116 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Abiud Wakhanu Mulongo. A Novel Divisibility Test for 7 Based on the Transformation N ′ = 4a − b. Authorea . 13 October 2025. DOI: https://doi.org/10.22541/au.176038359.95362796/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. 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