Integer which is Multiplied by 9 when moving Last Digit to First
Theorem
Let $N$ be the positive integer:
- $N = 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$
$N$ is the smallest positive integer $N$ such that if you move the last digit to the front, the result is the positive integer $9 N$.
Corollary
The positive integers formed by concatenating the decimal representation of $N$ with itself any number of times have the same property:
- $\sqbrk {NN}, \sqbrk {NNN}, \sqbrk {NNNN}, \ldots$
Proof 1
First we demonstrate that $N$ has this property:
\(\ds \) | \(\) | \(\ds 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719 \times 9\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 91 \, 011 \, 235 \, 955 \, 056 \, 179 \, 775 \, 280 \, 898 \, 876 \, 404 \, 494 \, 382 \, 022 \, 471\) |
A way to find $N$ is given below.
Suppose $N = \sqbrk {a_k a_{k - 1} \dots a_1}$ satisfies the property above.
Consider the rational number represented by $q = 0.\dot {a_k} a_{k - 1} \dots \dot {a_1}$.
Since $a_k \ne 0$, we require $q \ge 0.1$.
By the property we have:
- $9 q = 0 . \dot {a_1} a_k \dots \dot {a_2}$
Then:
- $90 q = a_1 . \dot {a_k} a_{k - 1} \dots \dot {a_1}$
Notice that the recurring part of $90 q$ is the same as the recurring part of $q$.
Subtracting, we get $89 q = a_1$.
We have:
- $a_1 \ge 89 \times 0.1 = 8.9$
so:
- $a_1 = 9$
which gives:
- $q = \dfrac 9 {89}$
Decimal Expansion
- $\dfrac 9 {89} = 0 \cdotp \dot 10112 \, 35955 \, 05617 \, 97752 \, 80898 \, 87640 \, 44943 \, 82022 \, 471 \dot 9$
$\Box$
which gives us:
- $N = 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$
as the smallest positive integer satisfying the property.
$\blacksquare$
Proof 2
From Integer which is Multiplied by Last Digit when moving Last Digit to First, $N$ is equal to the recurring part of the fraction:
- $q = \dfrac {a_1} {10 a_1 - 1}$
where $a_1 = 9$.
Thus:
- $q = \dfrac 9 {10 \times 9 - 1} = \dfrac 9 {89}$
Hence:
Decimal Expansion
- $\dfrac 9 {89} = 0 \cdotp \dot 10112 \, 35955 \, 05617 \, 97752 \, 80898 \, 87640 \, 44943 \, 82022 \, 471 \dot 9$
$\Box$
The result follows.
$\blacksquare$
Also see
- Integer which is Multiplied by Last Digit when moving Last Digit to First: the example given here is a special case
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10,112,359,550,561,797,752,808,988,764,044,943,820,224,719$