Henry Ernest Dudeney/Modern Puzzles/62 - Factorizing/Solution

Modern Puzzles by Henry Ernest Dudeney: $62$

Factorizing

What are the factors (the numbers that will divide it without any remainder) of this number -- $1000000000001$?

This is easily done if you happen to know something about numbers of this peculiar form.

In fact, it is just as easy for me to give two factors if you insert, say $101$ noughts, instead of $11$, between the two ones.

There is a curious, easy, and beautiful rule for these cases.

Can you find it?

Solution

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

\(=\)

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

\(\ds \)

\(=\)

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

Proof

This specific result is an instance of Divisors of One More than Power of 10: Number of Zero Digits Congruent to 2 Modulo 3.

Also see

• Divisors of One More than Power of 10
• Divisors of One More than Power of 10: Number of Zero Digits Even

• Prime Factors of One More than Power of 10

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