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&amp;^2&amp;^3&amp;^4&amp;^5&amp;^6 \\<br /> &amp;&amp;&amp; A&amp;B&amp;C \\<br /> \times &amp;&amp; 1 &amp; 0&amp; 0 &amp; 2 \\ \hline<br /> &amp;&amp;&amp; 2A &amp; 2B&amp; 2C \\<br /> A&amp;B&amp;C \\ \hline<br /> A&amp;B&amp;0&amp;0&amp;7&amp;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.