Calculating the Last Five Digits of 678910: A Modular Arithmetic Approach

Understanding and calculating the last five digits of large numbers can be crucial in various fields, including computer science, cryptography, and number theory. This article explains the process of finding the last five digits of 678910, using the principles of modular arithmetic, particularly in the context of modular arithmetic.

Introduction to Modular Arithmetic

Modular arithmetic, also known as clock arithmetic, is a system in which we only care about the remainder when one number is divided by another. It is the method behind finding the last few digits of a large number. The last five digits of a number are equivalent to finding its remainder when divided by 100,000 (i.e., mod 100,000).

Breaking Down the Calculation

To find the last five digits of 678910, we can use the following steps:

Step 1: Calculate 67892 Mod 100,000

First, we start by calculating (6789^2).

67892

Expanding it, we get:

67892  46072521

To find the last five digits, we take the remainder of 46072521 when divided by 100,000.
46072521 mod 100,000  12,080
Step 2: Calculate 67894 Mod 100,000
Next, we calculate (6789^4) using the result from the previous step.
(67892)(67892)  46072521 * 46072521
Breaking it down, we get:
46072521 * 46072521  212,100,524,080,121
Now we take the remainder of this result when divided by 100,000.
212,100,524,080,121 mod 100,000  250,021
Step 3: Calculate 67898 Mod 100,000
Using the result from the previous step, we now calculate (6789^8).
(67894)(67894)  250,021 * 250,021
Breaking it down, we get:
250,021 * 250,021  62,506,250,280,441
Now we take the remainder of this result when divided by 100,000.
62,506,250,280,441 mod 100,000  19,361   176,001
After simplifying, we get:
176,481 mod 100,000  76,481
Step 4: Calculate 678910 Mod 100,000
Finally, we use the results from the previous steps to calculate (6789^{10}).
First, split the exponent into 8 and 2:
678910  (67898)(67892)
Using the previous results:
(76,481)(12,080)
Breaking it down, we get:
76,481 * 12,080  921,896,520
Now we take the remainder of this result when divided by 100,000.
921,896,520 mod 100,000  36,601
Conclusion
The last five digits of 678910 are 36,601. This is achieved by breaking down each exponent and performing the necessary modular operations step-by-step. This method not only helps in solving the problem but also in understanding the principles of modular arithmetic, which are extensively used in various computational and theoretical contexts.
Keywords
Modular Arithmetic, Last Five Digits, 6789^10
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