Finding an unknown 3-digit number

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SUMMARY

The discussion centers on the impossibility of finding a 3-digit number ABC that, when multiplied by 1002, results in the format AB007C. Through analysis using the distributive property, it is established that C must equal 0, leading to a contradiction in the fifth column where 2B cannot end in 7. Therefore, the conclusion is that this alphametic problem has no valid solution.

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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!
 
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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:

[tex]\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}[/tex]

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

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

Q.E.D.

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

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

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