Properties of Periodic Part of Reciprocal of 31
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Theorem
We have from Reciprocal of $31$ that the decimal expansion of the reciprocal of $31$ is:
- $\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$
Then:
| \(\ds 032258 \times 2\) | \(=\) | \(\ds 64 \, 516\) | ||||||||||||
| \(\ds 032258 \times 4\) | \(=\) | \(\ds 129 \, 032\) | ||||||||||||
| \(\ds 032258 \times 5\) | \(=\) | \(\ds 161 \, 290\) | ||||||||||||
| \(\ds 032258 \times 7\) | \(=\) | \(\ds 225 \, 806\) | ||||||||||||
| \(\ds 032258 \times 8\) | \(=\) | \(\ds 258 \, 064\) | ||||||||||||
| \(\ds 032258 \times 9\) | \(=\) | \(\ds 290 \, 322\) | ||||||||||||
| \(\ds 032258 \times 14\) | \(=\) | \(\ds 451 \, 612\) | ||||||||||||
| \(\ds 032258 \times 16\) | \(=\) | \(\ds 516 \, 128\) | ||||||||||||
| \(\ds 032258 \times 18\) | \(=\) | \(\ds 580 \, 644\) | ||||||||||||
| \(\ds 032258 \times 19\) | \(=\) | \(\ds 612 \, 902\) |
| \(\ds 03225 + 80645 + 16129\) | \(=\) | \(\ds 99 \, 999\) | ||||||||||||
| \(\ds 032 + 258 + 065 + 416 + 129\) | \(=\) | \(\ds 900\) |
Proof
Verified by calculation.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $31$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $31$