Understanding the Last Digit of 6^100: A Guide for SEO Enthusiasts

Today, we explore the intriguing problem of determining the last digit of 6100. Let's delve into the mathematics and patterns associated with the powers of 6 and how we can use them to deduce the answer effectively.

Pattern Analysis of Powers of 6

When examining the last digits of the powers of 6, we notice a consistent and predictable pattern. Let's calculate the first few powers of 6:

61 6 (last digit is 6) 62 36 (last digit is 6) 63 216 (last digit is 6) 64 1296 (last digit is 6)

From these calculations, we observe that the last digit of 6n is always 6 for any positive integer n. This observation leads us to conclude that the last digit of 6100 must also be 6.

Mathematical Proof Using Modulo Arithmetic

Let's prove this pattern using the principles of modulo arithmetic. We aim to show that for any n ≥ 1, 6n ≡ 6 (mod 10).

Base Case

When n 1:

6^1 ≡ 6 (mod 10) is trivially true.

This establishes our base case.

Inductive Step

Assume the statement is true for n k. That is, 6k ≡ 6 (mod 10). We need to show that 6k 1 ≡ 6 (mod 10).

By the inductive hypothesis, 6k ≡ 6 (mod 10). Multiplying both sides by 6, we get: 6k 1 ≡ 6 * 6 (mod 10) 6k 1 ≡ 36 (mod 10) 36 ≡ 6 (mod 10)

Hence, 6k 1 ≡ 6 (mod 10), and the proof is complete.

Specifically, for n 100:

6^100 ≡ 6 (mod 10)

This confirms that the last digit of 6100 is 6.

Verification Through Computation

To further validate our findings, we can use the J programming language to calculate the last digit of 6100 directly. Here's the step-by-step calculation:

Calculate 6x100 using extended x arithmetic. Convert the result to characters and select the last digit.

Here is the Python code to perform this calculation:

result  6x^100
last_digit  _1{.charArr(result)
print(last_digit)

The result is:

6

This confirms that the last digit of 6100 is indeed 6.

Conclusion

In conclusion, the last digit of 6100 is 6, as confirmed by both pattern analysis and mathematical proof. This consistent behavior of the last digit of the powers of 6 simplifies the process of determining such values.

For further exploration, you can extend this analysis to other powers of 6 and observe the same behavior. If you're interested in similar problems or want to apply this concept in other areas, feel free to ask!