Find four digit $abcd$ such that $ab \cdot cd = abcd$ I stumbled upon this problem and I can't seem to figure it out. It's an easy-looking problem and it goes like  this: 

Find the $4$-digit number $abcd$ such that if you multiply the two
  $2$-digit numbers $ab$ and $cd$, you again get the $4$-digit number
  $abcd$.

I started like this: 
Let $x = 10a + b$ and $y = 10 c + d$, with $a,b,c,d \in \{0,\ldots, 9\}$. The problem is then reduced to: $x\cdot y = (10a + b)(10c+d) = 1000a + 100b + 10c + d$, with the latter being the $4$-digit number $abcd$. 
Or equivalently, $$100ac + 10 ad + 10bc + bd = 1000a + 100b + 10 c + d.$$ Unfortunately, this reasoning doesn't get me very far. Any ideas?
 A: That's impossible!
For $ab * cd = abcd$, $cd$ would need to be at least $100$
A: Take your equation
$$100ac + 10 ad + 10bc + bd = 1000a + 100b + 10 c + d$$
and solve for $d$:
$$d = -\frac{10\left(bc+10ac-c-10b-100a\right)}{b+10a-1}.$$
Dividing the numerator by the denominator leaves a remainder of $100$, so $(10a+b-1) \mid 100$. You can take it from there, I bet.
A: Hint: I'll use the line to indicate the base-10 representation of the number rather than the product. Consider the thousands digit: on the left-hand side, you know it is equal to the first digit of $\overline{ac}$, plus a possible carry. This should help with some of the case work.
Additionally, taking both sides modulo 5 tells you something helpful about $b$ as well.
Edit: Missed an easy one: $100(\overline{ab}) = \overline{ab00} \leq \overline{abcd}$. So $\frac{\overline{abcd}}{\overline{ab}}$ can't have 2 digits. In fact, we only have equality in the case where $\overline{cd} = 00$, which gives the trivial solution $0000$.
A: I don't think this problem has a solution: 
consider the 2 positive integers  $x,y$ that satisfy: 
$$
100x + y = xy \\
100x = y(x-1)
$$
looking at the above equation you can see that $x|y(x-1)$ but $x$ and $x-1$ are coprime so $x|y \Rightarrow y=kx \: k \in \mathbb{N}$ and I can rewrite the equations and get: 
$$
100=k(x-1) \\
y=kx \\
\therefore y>100
$$
this is a contradiction since $y$ has 2 digits.
