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

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.