Since \( z \) is divisible by 7, 11, and 13, compute the least common multiple:
["Finding the Least Common Multiple of 7, 11, and 13: A Quick Guide to Divisible Numbers", "When a number ( z ) is divisible by 7, 11, and 13, determining its least common multiple (LCM) becomes straightforward thanks to fundamental math principles. Since these three numbers—7, 11, and 13—are all prime, their least common multiple is simply their product.", "### Why the LCM of 7, 11, and 13 is Their Product", "A least common multiple is the smallest positive integer divisible by each of the given numbers. When numbers are all prime and share no common factors other than 1, their LCM is just the multiplication of all the numbers together.", "#### Step-by-Step LCM Calculation", "Given:\n- ( z ) divisible by 7, 11, and 13\n- All three are prime numbers", "Calculate:\n[\n\ ext{LCM}(7, 11, 13) = 7 \ imes 11 \ imes 13\n]", "Now compute the product:", "[\n7 \ imes 11 = 77\n]\n[\n77 \ imes 13 = 1001\n]", "Thus,\n[\n\ ext{LCM}(7, 11, 13) = 1001\n]", "### Why This Matters", "The number 1001 appears frequently in number theory and real-world applications, such as calendar cycles and cryptography, precisely because it arises naturally as the smallest common multiple of these primes. Understanding the LCM of prime numbers helps build a strong foundation for more complex number patterns and divisibility rules.", "---", "Summary\nSince 7, 11, and 13 are all prime and pairwise coprime, their least common multiple is their product:\n[\n\boxed{1001}\n]\nThis number is the smallest positive integer divisible by 7, 11, and 13."]
