Solution: We are to find the smallest positive integer $ n $ such that:
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- Since $ k $ must be a non-negative integer, valid values are $ k = 0, 1, 2, ..., 13 $, which gives 14 values.
- Thus, there are $ \boxed{14} $ integers between 1 and 100 that are congruent to 3 modulo 7.
- Question: What is the smallest positive integer $ n $ such that $ n^2 \equiv 1 \pmod{12} $?
- n^2 \equiv 1 \pmod{12}
- This implies $ n^2 - 1 \equiv 0 \pmod{12} $, or $ (n - 1)(n + 1) \equiv 0 \pmod{12} $.
- So $ 12 \mid (n - 1)(n + 1) $.
