The Lucas sequence is the sequence 1, 3, 4, 7, 11, $\ldots$ where the first term is 1, the second term is 3 and each term after that is the sum of the previous two terms. What is the remainder when the $100^{\mathrm{th}}$ term of the sequence is divided by 8?
I start by listing the first few terms of the Lucas sequence and their remainders when divided by 8: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, $\ldots$ and 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, $\ldots$ I notice that the remainders repeat every 12 terms, and the cycle is 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2. This means that the remainder of the $n^{\mathrm{th}}$ term depends on the value of $n$ modulo 12. So, to find the remainder of the $100^{\mathrm{th}}$ term, I need to find the value of 100 modulo 12. I can use long division or a calculator to find that 100 divided by 12 gives a quotient of 8 and a remainder of 4. Therefore, the $100^{\mathrm{th}}$ term has the same remainder as the 4th term, which is 7. 