If it is 15h now, what time will it be in 10 hours? It’s not “25 h”, but 1 am: on the clock, after 24 the count starts again. Gauss gave a name and symbol to this idea in Disquisitiones:

$$25 \equiv 1 \pmod{24}$$

It reads “25 is congruous to 1, modulo 24”: both leave the same remainder when divided by 24. The symbol $\equiv$ works almost like an equals sign: you can add and multiply on both sides.

Application: the nines-out. As $10 = 9 + 1$, we have $10 \equiv 1 \pmod 9$, and then $100 \equiv 1$, $1000 \equiv 1$, and so on. That’s why:

Why Nines-Outs Work (Click to Next)

  1. Write the number in decimal places: $4528 = 4 \cdot 1000 + 5 \cdot 100 + 2 \cdot 10 + 8$.
  2. Replace each power of 10 with 1, which has the same remainder when divided by 9: $4528 \equiv 4 + 5 + 2 + 8 = 19 \pmod 9$.
  3. Repeat with 19: $19 \equiv 1 + 9 = 10 \equiv 1 + 0 = 1 \pmod 9$.
  4. Conclusion: 4528 leaves remainder 1 when dividing by 9. Check: $4528 = 9 \cdot 503 + 1$.
Enter any number: the sum of the digits, repeated until only one remains, gives the remainder when divided by 9.

It’s the famous “test of nines” that was learned at school to check multiplications. Nas Disquisitiones, Gauss also proved the law of quadratic reciprocity, which he called the theorem aureum, the golden theorem. He liked it so much that he found eight different demonstrations.

All topics by Carl Friedrich Gauss · ← The polygon with 17 sides · The fundamental theorem of algebra →

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