what is the 100 and 300 number in this sequence 3279163279163279

To find the 100th and 300th numbers in the repeating sequence \(3279163279163279\), we need to understand the structure of the sequence. 

The sequence \(3279163279163279\) has a repeating cycle with a length of 16 characters. This means every 16 characters, the sequence starts over from the beginning.

1. **Find the 100th number:**

   To determine the position within the cycle for the 100th number, we use:
   \[
   100 \mod 16
   \]
   \[
   100 \div 16 = 6 \text{ remainder } 4
   \]
   Therefore:
   \[
   100 \mod 16 = 4
   \]
   So, the 100th number corresponds to the 4th position in the repeating cycle \(3279163279163279\).

   Checking the sequence, the 4th number is \(9\).

2. **Find the 300th number:**

   To determine the position within the cycle for the 300th number, we use:
   \[
   300 \mod 16
   \]
   \[
   300 \div 16 = 18 \text{ remainder } 12
   \]
   Therefore:
   \[
   300 \mod 16 = 12
   \]
   So, the 300th number corresponds to the 12th position in the repeating cycle \(3279163279163279\).

   Checking the sequence, the 12th number is \(6\).

Thus:
- The 100th number in the sequence is \(9\).
- The 300th number in the sequence is \(6\).

Would you like more details or have any questions?

Here are 8 related questions to consider:

1. How can you generalize finding any \(n\)-th number in a repeating sequence?
2. What is the importance of modular arithmetic in sequences?
3. How would you describe a sequence that repeats after a certain number of terms?
4. Can you find the 200th number in this sequence?
5. What is the difference between a finite and infinite sequence?
6. How do periodic sequences appear in real-world applications?
7. How can sequences be represented graphically?
8. What are some techniques to identify the period of a sequence?

**Tip:** When dealing with repeating sequences, always check the length of the cycle and use modular arithmetic to find the position within the cycle.

Learn how to find specific numbers in a repeating sequence (3279163279163279). Understand the use of modular arithmetic to determine positions within the sequence cycle. Suitable for students exploring sequences and modular arithmetic.

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