The 'Double & Subtract' Divisibility by 7 Trick

Ever get stuck wondering if a large number divides perfectly by 7? Forget long division! There's a lightning-fast trick!

Subject: number-theory • Classes: 6–12 • Difficulty: intermediate

The Trick

To check if a number is divisible by 7, take its last digit, double it, and subtract this result from the remaining part of the number. If the new, smaller number is divisible by 7 (or 0), then the original number is also divisible by 7. Repeat if needed! **Why it works:** Consider $N = 10a + b$. We test $a - 2b$. Observe: $10a + b = 10(a - 2b) + 21b$. Since $21b$ is clearly divisible by 7, $N$ is divisible by 7 if and only if $10(a - 2b)$ is divisible by 7. As 10 and 7 are coprime, this means $a - 2b$ must be divisible by 7. Genius!

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