\(4321012345 \times 98766789\)?
Finding the Last Three Digits of a Large Multiplication
The question asks for the last three digits of the product obtained when multiplying \(4321012345\) by \(98766789\). To find the last three digits of any number, we need to determine the remainder when that number is divided by \(1000\). This is also known as finding the number modulo \(1000\).
A key property of multiplication is that the last three digits of the product depend only on the last three digits of the numbers being multiplied. We can express this using modular arithmetic:
\((a \times b) \pmod{1000} = ((a \pmod{1000}) \times (b \pmod{1000})) \pmod{1000}\)
Step-by-Step Calculation
-
Identify the last three digits of each number:
- The last three digits of \(4321012345\) are \(345\). So, \(4321012345 \pmod{1000} = 345\).
- The last three digits of \(98766789\) are \(789\). So, \(98766789 \pmod{1000} = 789\).
-
Multiply these last three digits:
We need to calculate the product of \(345\) and \(789\).
\(345 \times 789\)
Let's perform the multiplication:
345 x 789 ------- 3105 (\(345 \times 9\)) 27600 (\(345 \times 80\)) 241500 (\(345 \times 700\)) ------- 272205
The result of the multiplication is \(272205\).
-
Find the last three digits of the result:
To find the last three digits of the product \(4321012345 \times 98766789\), we take the last three digits of \(272205\).
\(272205 \pmod{1000} = 205\)
The last three digits are \(2\), \(0\), and \(5\).
Conclusion
Therefore, the last three digits in the multiplication of \(4321012345 \times 98766789\) are \(2\), \(0\), and \(5\).
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