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.