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

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

\(\ds 1001\)

\(=\)

\(\ds 11 \times 91\)

\(\ds 1 \, 000 \, 001\)

\(=\)

\(\ds 101 \times 9901\)

\(\ds 1 \, 000 \, 000 \, 001\)

\(=\)

\(\ds 1001 \times 999 \, 001\)

\(\ds \)

\(=\)

\(\ds 7 \times 11 \times 13 \times 19 \times 52 \, 579\)

\(\ds 1 \, 000 \, 000 \, 000 \, 001\)

\(=\)

\(\ds 10 \, 001 \times 99 \, 990 \, 001\)

\(\ds \)

\(=\)

\(\ds 73 \times 137 \times 99 \, 990 \, 001\)

$101$ Zero Digits

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

\(=\)

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

Demonstration

By long multiplication:

     99990001
    x   10001
     --------
     99990001
 999900010000
-------------
1000000000001

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