The Lucky 7 Divisibility Trick!

Struggling to figure out if a big number is divisible by 7 without a calculator? Forget long division! There's a quick, magical trick that makes 7's divisibility surprisingly easy!

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

The Trick

Here's the trick: Take any number. Cut off its last digit. Double that last digit. Now, subtract this doubled value from the 'remaining' part of your original number. If the new number you get is 0, 7, or a multiple of 7, then your original number is divisible by 7! You can repeat this process for larger numbers until you get a small, recognizable number. \textbf{Why it works:} Let the number be $10a + b$. We perform $a - 2b$. If $a-2b$ is a multiple of 7, say $7k$, then $a = 7k+2b$. Substituting back into the original number: $10(7k+2b) + b = 70k + 20b + b = 70k + 21b = 7(10k + 3b)$, which is clearly a multiple of 7!

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