Midy's Theorem/Examples/7
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Example of Use of Midy's Theorem
Let $p = 7$.
We have for $a = 1$ and $b = 10$ the decimal expansion:
- $\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$
Hence:
- $N = 142857$
This means that:
- $\alpha = 6 = 2 \times 3$
Midy's Theorem states that $N$ is divisible by $10^2 - 1$ and $10^3 - 1$.
Moreover, we can partition $N$ into blocks of digits of equal length:
- $N = 14 \times 100^2 + 28 \times 100 + 57$
- $N = 142 \times 1000 + 857$
Summing these blocks together, we obtain:
- $14 + 28 + 57 = 99 = 10^2 - 1$
and:
- $142 + 857 = 999 = 10^3 - 1$