The '7' Divisibility Detector!
Ever struggled to check if a large number is divisible by 7 without a calculator? No more guessing games!
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 it from the rest of the number. If the new number is divisible by 7 (or 0), then the original number is too! Repeat if the number is still large. \n\nWHY it works: Let the number be $10a + b$. If $10a + b$ is divisible by 7, then $10a + b \equiv 0 \pmod{7}$. Multiplying by 5 (the modular inverse of 10 mod 7), we get $5(10a+b) \equiv 0 \pmod{7}$, which simplifies to $50a + 5b \equiv 0 \pmod{7}$. Since $50 \equiv 1 \pmod{7}$ and $5 \equiv -2 \pmod{7}$, this becomes $a - 2b \equiv 0 \pmod{7}$. Thus, if $a-2b$ is divisible by 7, so is $10a+b$!
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