Finding an unknown 3-digit number

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In summary, the conversation discusses the problem of finding the original 3-digit number when it is multiplied by 1002 and results in a specific 6-digit number. The conversation mentions trying different methods but no solution has been found yet. The expert then provides a way to arrange the multiplication and concludes that the problem has no solution.
  • #1
mklilly2000
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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 teh 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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  • #2
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}
^1&^2&^3&^4&^5&^6 \\
&&& A&B&C \\
\times && 1 & 0& 0 & 2 \\ \hline
&&& 2A & 2B& 2C \\
A&B&C \\ \hline
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.

 
  • #3
Soroban, the problem does not say that the original number is ABC.
 
  • #4
Hint:

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

FAQ: Finding an unknown 3-digit number

What is the process for finding an unknown 3-digit number?

The process for finding an unknown 3-digit number involves using clues and logic to narrow down the possibilities. This can be done through mathematical equations, patterns, or trial and error.

Can you give an example of how to find an unknown 3-digit number?

Sure, for example, if the sum of the digits is 15 and the hundreds digit is double the tens digit, we can create the equation 2x + y + z = 15, where x is the hundreds digit, y is the tens digit, and z is the ones digit. From there, we can try different combinations of numbers that fit the equation until we find the correct answer, such as 6 + 6 + 3 = 15, making the unknown 3-digit number 663.

Are there any shortcuts or tricks for finding an unknown 3-digit number?

There are some common patterns and tricks that can be used to find an unknown 3-digit number more quickly. For example, if the last two digits of the number are the same, the number must be divisible by 11. Additionally, if the digits add up to a multiple of 3, then the number itself must be a multiple of 3.

Is there a specific method or strategy that is most effective for finding an unknown 3-digit number?

The most effective method for finding an unknown 3-digit number will depend on the specific clues and information given. It is important to carefully analyze the information and use critical thinking skills to determine the best approach. Some people may find success with trial and error, while others may prefer using equations and mathematical reasoning.

Can finding an unknown 3-digit number be applied to real-world situations?

Yes, the process of finding an unknown 3-digit number can be applied to various real-world situations, such as solving mathematical equations, decoding secret codes, or even determining passwords or combinations. It is a valuable skill that can be used in many different contexts.

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