Henry Ernest Dudeney/Puzzles and Curious Problems/172 - Curious Multiplicand/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $172$

Curious Multiplicand
What number is it that can be multiplied by $1$, $2$, $3$, $4$, $5$, or $6$ and no new figures appear in the result?


Solution

$142 \, 857$


Proof

This is, of course, the recurring part of the decimal expansion of the Reciprocal of 7:

$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$


Hence:

\(\ds 1 \times 142 \, 857\) \(=\) \(\ds 142 \, 857\)
\(\ds 2 \times 142 \, 857\) \(=\) \(\ds 285 \, 714\)
\(\ds 3 \times 142 \, 857\) \(=\) \(\ds 428 \, 571\)
\(\ds 4 \times 142 \, 857\) \(=\) \(\ds 571 \, 428\)
\(\ds 5 \times 142 \, 857\) \(=\) \(\ds 714 \, 285\)
\(\ds 6 \times 142 \, 857\) \(=\) \(\ds 857 \, 142\)

$\blacksquare$


Also see


Sources

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