Finding an unknown 3-digit number

[mklilly2000]

If you multiply a 3-digit number by 1002 and get AB007C, where A, B, and C stand for digits, what is the original 3-digit number? We've tried using the distributive property as well as just multiplying by 3-digit numbers that give you a 0 in the thousandths place but no luck. Thanks for any suggestions!

 

[soroban]

Hello, mklilly2000!

This alphametic has no solution.

If you multiply a 3-digit number ABC by 1002 and get AB007C,
what is the original 3-digit number?

Arrange the multiplication like this:

$$\begin{array}{ccccc}<br /> ^1&^2&^3&^4&^5&^6 \\<br /> &&& A&B&C \\<br /> \times && 1 & 0& 0 & 2 \\ \hline<br /> &&& 2A & 2B& 2C \\<br /> A&B&C \\ \hline<br /> A&B&0&0&7&C \end{array}$$

In column-6, we see that $$2C$$ ends in $$C.$$
Hence, $$C=0$$ and there is no "carry" to column-5.

In column-5, we see that $$2B$$ ends in 7.
Since $$2B$$ is even, this is clearly impossible.

Q.E.D.

 

[Evgeny.Makarov]

Soroban, the problem does not say that the original number is ABC.

 

[Greg]

Hint:

Spoiler

The middle digit of the three-digit number is a 3.