# Puzzle: concatenation is three times product

The numbers A and B have three digits, while C is an odd number with 5 digits. Say you were asked to calculate the integer $(A B) /C$. Instead though you put A and B next to each other to form a 6-digit number D, and then divided by C. Your answer is now three times the correct answer. Find A, B and C.

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Right so we are trying to solve the equation:

$\frac{3AB}{C} = \frac{1000A + B}{C}$

We can rearrange and factorise this to give:

$(3A - 1)(3B - 1000) = 1000$

Now $3A - 1 \geq 299$ and is an integer factor of $1000$. The only possibilities for this number are $500$ and $1000$. It is easy to see that we must have the first of these, hence $A = \frac{501}{3} = 167$.

Then $3B - 1000 = 2$, giving $B = \frac{1002}{3} = 334$.

To find $C$ remember that it has $5$ digits and is odd. It also divides $AB$.

Now $AB = 55778 = 2\times 167^2$ so that the only possibility for $C$ is $167^2 = 27889$

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