Divisors of One More than Power of 10/Number of Zero Digits Congruent to 2 Modulo 3/Examples/101 Zero Digits

Example of Divisors of One More than Power of 10: Number of Zero Digits Congruent to 2 Modulo 3

\(\ds 1 \underbrace {00 \ldots 0}_{\text {$101$ zeroes} } 1\)

\(=\)

\(\ds 1 \underbrace {00 \ldots 0}_{\text {$33$ zeroes} } 1 \times \underbrace {99 \ldots 9}_{\text {$34$ $9$s} } \underbrace {00 \ldots 0}_{\text {$33$ zeroes} } 1\)

Proof

$101$ is of the form $3 \times k - 1$ where $k = 34$.

The result follows directly from Divisors of One More than Power of 10: Number of Zero Digits Congruent to 2 Modulo 3.

$\blacksquare$

Also see

• Henry Ernest Dudeney: Modern Puzzles: $62$ -- Factorizing

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $62$. -- Factorizing
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $113$. Factorizing