Finding last digits of 983389^389

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In summary, the problem is to reduce the base of a discrete mathematics problem, specifically in the number $983,389^{389}$, to $389$ (mod 1000) in order to find the last three digits of the exponent. This step is necessary in order to solve the problem and move on to step 3.
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Alexthexela
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Posted is a discrete mathematics problem. I'm having trouble with step 2, where I'm instructed to "reduce the base." Does this refer to the logarithmic base? I'm looking through my textbook and at help articles online, but still finding myself confused. I'm new to this type of problem and seeking advice on this spot in particular, but any guidance you can provide would be much appreciated. Thank you all.View attachment 8063
 

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Alexthexela said:
Posted is a discrete mathematics problem. I'm having trouble with step 2, where I'm instructed to "reduce the base." Does this refer to the logarithmic base? I'm looking through my textbook and at help articles online, but still finding myself confused. I'm new to this type of problem and seeking advice on this spot in particular, but any guidance you can provide would be much appreciated. Thank you all.
Hi Alexthexela, and welcome to MHB!

In the number $983,389^{389}$, the base is $983,389$ and the exponent is $389$. The problem tells you to "think (mod 1000)". To do that, the first step is to "reduce the base" (mod 1000), in other words to replace $983,389$ by $389$. The reason for doing that is that the last three digits of $389^{389}$ will be the same as the last three digits of $983,389^{389}$. That is all there is to step 2 of the problem, and you can then move on to step 3.
 

FAQ: Finding last digits of 983389^389

1. How do you find the last digit of a large number like 983389^389?

To find the last digit of a large number, we can use modular arithmetic. This means we divide the number by a certain value and look at the remainder. In this case, we can use the pattern of the last digit to simplify the calculation.

2. Can you explain the pattern of the last digit in powers of 9?

When we raise 9 to any power, the last digit follows a repeating pattern: 9, 1, 9, 1, and so on. This means that the last digit of 983389^389 will also be 1.

3. Is there a specific method to find the last digit of any number raised to a power?

There are certain patterns and rules that can be used to find the last digit of any number raised to a power. For example, the last digit of any number raised to an even power will be the same as the last digit of the original number. It is helpful to have a strong understanding of modular arithmetic to use these methods effectively.

4. What is the significance of finding the last digit of a large number?

Knowing the last digit of a large number can be helpful in verifying calculations or identifying patterns in numbers. It can also be useful in cryptography and other mathematical applications.

5. Are there any other techniques for finding the last digit of a large number raised to a power?

Yes, there are other techniques such as using a calculator or writing out the full calculation. However, these methods can be time-consuming and prone to errors. Using modular arithmetic and understanding the patterns of numbers can be a more efficient and accurate way to find the last digit.

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