Divisibility 1,2,3,4,5,6,7,8,9,&10
Solution 1:
Suppose the number is of form $N=jihgfedcba$. We may write:
$a+b+c+d+e+f+g+h+i+j+(10-1)b+(10^2-1)c+(10^3-1)d+(10^4-1)e+(10^5-1)f+(10^6-1)g+(10^7-1)h+(10^8-1)i+(10^9-1)j$
A: Any number such as following forms are divisible by 2, 4, 5 and 8:
$(2k)(40)$ such as $240, 440, 640, 840 . . .$
$(2k+1)(20)$, such as $120, 320, 520, 720, . . .$
B: Whatever the value of g is, the term $\frac{10^6-1}{9}$ is divisible by $77$.
C: For 7 we consider the remainder of $10^n-1$ when divided by 7:
$T=.....10, 10^2, 10^3, 10^4, 10^5, 10^6, 10^7, 10^8, 10^9$
$R_{10^n}=...3,..2,.. 6,.. 4,.. 5,... 1,.. 3,.. 2,.. 6$
$R_{10^n-1}=2,..1,..5,..3,...4,...0,...2,..1,..5$
We can make following relation for divisibility by 7:
$a+b+c+d+e+f+g+h+i+j+(2)b+(1)c+(5)d+(3)e+(4)f+(0)g+(2)h+(1)i+(5)j≡ mod 7$
D: For 11 we just consider the remainder of $10^n-1$ for odd n because for even n,. $(10^n-1)$ is divisible by 11 :
$T= .......10,....10^3,...10^5,...10^7,..10^9$
$R_{10^n}=...-1,...-1,...-1,...-1,..-1$
$R_{10^n-1}=-2,...-2,...-2,...-2,..-2$
So we can make following relation for divisibility by 11:
$a+b+c+d+e+f+g+h+i+j+(-2)b+(-2)d+(-2)f+(-2)h+(-2)j≡ mod 11$
So we have following system of Diophantine equations:
$a+b+c+d+e+f+g+h+i+j+(-2)b+(-2)d+(-2)f+(-2)h+(-2)j≡ mod 11$
$a+b+c+d+e+f+g+h+i+j+(2)b+(1)c+(5)d+(3)e+(4)f+(0)g+(2)h+(1)i+(5)j≡ mod 7$
The sum $a+b+c+. . .i+j= \frac{9(9+1)}{2}=45$ is divisible by 3 and 9. This system of equations indicates that the question can have numerous solutions, to find one for example take $cba=840$ which is divisible by 2, 3, 4, 5, 7 and 8, That is we assume $a=0$, $b=4$ and $c=8$ and look for other digits as follows, we have:
$45+4\times2+8\times1+5d+3e+4f+2h+i+5j≡ mod 7$
Or:
$61+5d+3e+4f+2h+i+5j≡ mod 7$
$45-2\times 4-2(d+f+h+j)=37-2(d+f+h+j)≡ mod 11$
Suppose $37-2(d+f+h+j)=11$ ⇒$d+f+h+j=(37-11)/2=13$
Suppose $d=1, . f=2,.h=3,. and,..j=7$ then we have:
$61+5+3e+8+6+i+35=115+3e+i≡ mod 7$
Let $115+3e+i=21\times 7=$⇒ $3e+i=32$ ⇒ $e=9$ and $i=5$
The only number which remains is 6 for g, so one solution can be:
$N=7536291840$
Solution 2:
If the digit representation of such number is $ \langle d_9 d_8 d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0 \rangle$, where $$ \langle d_9 d_8 d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0 \rangle:=\Sigma_{i=0}^{9}10^id_i$$ then we know that $d_0=0$ because $$10\mid \langle d_9 d_8 d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0 \rangle.$$
The sum $$d_9 +d_8+ d_7+ d_6+ d_5+ d_4+ d_3+ d_2+ d_1 +0$$ is $$0+1+2+3+4+5+6+7+8+9=45,$$ so $$9 | \langle d_9 d_8 d_7 d_6 d_5 d_4 d_3 d_2 d_1 0\rangle.$$
Because $$8\mid \langle a_2 a_10\rangle.$$ we also know that $$4\mid \langle a_2a_1\rangle \tag{2}$$
So we have $$t\mid \langle d_9 d_8 d_7 d_6 d_5 d_4 d_3 d_2 d_1 0 \rangle, \; \forall t \in \{2,3,4,5,6,8, 9, 10\}$$ if $(2)$ holds.
If
$$11\mid \langle d_9 d_8 d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0 \rangle$$
then for the alternate sum holds
$$11 \mid d_9 +d_8- d_7+ d_6- d_5+ d_4- d_3+ d_2- d_1+0$$
The alternate sum is between $$-9-8-7-6-5+1+2+3+4=35$$ and $$9+8+7+6-1-2-3-4-5=30.$$ But we know that the alternate sum is divisible by $11$ and it is the sum of $5$ odd and $5$ even numbers, so it is odd. Therefore the alternate sum is in $ \{-33,-11,11\}.$
How to construct a solution?
- Set $a_0=0$
-
Start with a possible value for $\langle a_2a_1\rangle$ such that
- $4\mid \langle a_2a_1\rangle$
- $a_1 \ne a_2$
- $a_1 \ne 0$
- $a_2 \ne 0$
The remaining digits $\{1,2,3,4,5,6,7,8,9\}\setminus\{d_1, d_2\}$ partion into two sets, set $\cal{O}=\{d_3, d_5, d_7, d_9\}$ that contains $4$ elements at the odd indexed positions and set $\cal{E}\{d_4, d_6, d_0\}$ that contains the $3$ elements at the even indexed positions.
- If $$\Sigma_{d \in \cal{E}}-\Sigma_{d \in \cal{O}}+a_2-a_1 \in \{-33,-11,11\} \tag{1}$$ we are done, otherwise select an element $\cal{e} \in \cal{E}$ and $\cal{o} \in \cal{O}$, remove $\cal{e}$ from $ \cal{E}$ and $\cal{o}$ from $ \cal{O}$ and add $\cal{e}$ to $ \cal{O}$ and $\cal{o}$ to$ \cal{E}.$ The left hand side of $(1)$ is incremented by $2(\cal{o}-\cal{e}).$ Repeat this step until $(1)$ is satisfied or if you are bored.
- If $(1)$ holds then assign the elements of $ \cal{O}$ to $d_9, d_7, d_5, d_3$ in an arbitrary way and assign the elements of $ \cal{E}$ to $a_8, a_6, a_4$ also in an arbitrary way. Now $$t\mid \langle d_9 d_8 d_7 d_6 d_5 d_4 d_3 d_2 d_1 0 \rangle, \; \forall t \in \{2,3,4,5,6,8, 9, 10,11\}$$ holds.
Example:
The smallest valid value for $\langle a_2 a_1 \rangle$ is $12.$ The values $00, 04, 08, 20$ are not valid because they contain $d_0.$ The numbers $44$ and $88$ are not valid because $d_2=d_1$, so the number cannot be a permutation.
So $d_1=6$ and $d_2=1$, we set $ \cal{O}=\{2,3,4,5\}$ and $ \cal{E}=\{7,8,9\}$. The left hand side of $(1)$ is $-14+24+1-6=-5.$ Now we shift $7$ to $\cal{O}$ and $4$ to $\cal{E}.$ This decreases the LHS of $(1)$ by $6$ to $-11$ and we are done. So we have $$\cal{O}=\{2,3,5,7\}$$ $$\cal{E}=\{4,8,9\}$$ $$\langle d_2 d_1 d_0 \rangle =160 $$ and can construct the number $$ 2435879160$$ $\square$
We can generate $4!\cdot 3!=144$ different numbers from our sets $\cal{O}$ and $\cal{E}.$ There is a good chance that about $\frac{1}{7}$ of these 144 numbers are divisible by $7$, these are about $20$ numbers. If there is no such number we can construct other numbers by repeating steps 2 to 5.
Here number $ 2435879160$ is already divisible by $7.$
Solution 3:
We just need a number divisible by $5,7,8,9,11$ and everything else is automatic. Divisibility by $9$ isn't a concern, as the digits already sum to $45$ and $9\mid45$. The number must end with $0$ since it's even and divisible by $5$. The last three digits must be divisible by $8$, so they're some multiple of $040$ (of course $040$ isn't actually a valid candidate, since it repeats $0$ twice). Divisibility by $11$ means the alternating sum of the digits must be a multiple of $11$. Divisibility by $7$ means the first $9$ digits is a multiple of $7$. Now, I think the most efficient method is to write a program taking into account all of these parameters to find a numerical answer.