Divisors of One More than Power of 10
Jump to navigation
Jump to search
Theorem
Prime Factors
\(\ds 11\) | \(=\) | \(\ds 11\) | ||||||||||||
\(\ds 101\) | \(=\) | \(\ds 101\) | ||||||||||||
\(\ds 1001\) | \(=\) | \(\ds 7 \times 11 \times 13\) | ||||||||||||
\(\ds 10 \, 001\) | \(=\) | \(\ds 73 \times 137\) | ||||||||||||
\(\ds 100 \, 001\) | \(=\) | \(\ds 11 \times 9091\) | ||||||||||||
\(\ds 1 \, 000 \, 001\) | \(=\) | \(\ds 101 \times 9901\) | ||||||||||||
\(\ds 10 \, 000 \, 001\) | \(=\) | \(\ds 11 \times 909 \, 091\) | ||||||||||||
\(\ds 100 \, 000 \, 001\) | \(=\) | \(\ds 17 \times 5 \, 882 \, 353\) | ||||||||||||
\(\ds 1 \, 000 \, 000 \, 001\) | \(=\) | \(\ds 7 \times 11 \times 13 \times 19 \times 52 \, 579\) | ||||||||||||
\(\ds 10 \, 000 \, 000 \, 001\) | \(=\) | \(\ds 101 \times 3541 \times 27961\) | ||||||||||||
\(\ds 100 \, 000 \, 000 \, 001\) | \(=\) | \(\ds 11^2 \times 23 \times 4093 \times 8779\) | ||||||||||||
\(\ds 1 \, 000 \, 000 \, 000 \, 001\) | \(=\) | \(\ds 73 \times 137 \times 99 \, 990 \, 001\) |
Number of Zero Digits Even
Let $N$ be a natural number of the form:
- $N = 1 \underbrace {000 \ldots 0}_{\text {$2 k$ $0$'s} } 1$
that is, where the number of zero digits between the two $1$ digits is even.
Then $N$ can be expressed as:
- $N = 11 \times \underbrace {9090 \ldots 90}_{\text {$k - 1$ $90$'s} } 91$
Number of Zero Digits Congruent to 2 Modulo 3
Let $N$ be a natural number of the form:
- $N = 1000 \ldots 01$
where the number of zero digits between the two $1$ digits is of the form $3 k - 1$.
Then $N$ has divisors:
- $1 \underbrace {00 \ldots 0}_{\text {$k - 1$ $0$'s} } 1$
- where the number of zero digits between the two $1$ digits is $k - 1$
- $\underbrace {99 \ldots 9}_{\text {$k$ $9$'s} } \underbrace {00 \ldots 0}_{\text {$k - 1$ $0$'s} }1$