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  1. Prime triplets:
    1. Let \(p\) be a natural integer. Show that exactly one of the three numbers \(p\), \((p + 10)\), and \((p + 20)\) is divisible by 3.
    2. Let \(a, b,\) and \(c\) be three natural integers representing the first three terms of an arithmetic progression with a common difference of 10. Determine these three numbers, knowing that they are all prime.
  2. Let \(E\) be the set of triplets of integers \((u; v; w)\) such that: $$3u + 13v + 23w = 0$$
    1. Show that for any such triplet, \(v \equiv w \pmod{3}\).
    2. We set \(v = 3k + r\) and \(w = 3k' + r\) where \(k, k',\) and \(r\) are integers and \(0 \le r \le 2\). Show that the elements of \(E\) are of the form: $$(-13k - 23k' - 12r, 3k + r, 3k' + r)$$

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