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📰 Check powers: is it divisible by \( 16 \)? Try 1–5: product \( 120 \), divisible by 8 but not 16. So 16 not guaranteed. 📰 Check 9: need two multiples of 3, but in five consecutive, only one multiple of 3 unless spaced correctly (e.g., 3 and 6: 6 is divisible by 3, 3 by 3 → only one multiple of 3 unless 9 appears). Sequence 7–11: none divisible by 3 → product not divisible by 3? Wait: 9 is divisible by 3, but if no multiple of 3, like 4–8: no multiple of 3 → product \( 4 \times 5 \times 6 \times 7 \times 8 \)? Wait, 6 is included → divisible by 3. 📰 Any five consecutive integers must include at least one multiple of 3? No: e.g., 1–5: no multiple of 3? 1,2,3,4,5 → 3 is there. 4–8: 6 → yes. 7–11: 9 → yes. 8–12: 9 and 12 → yes. Actually: in any 5 consecutive integers, the distance is less than 6, so by pigeonhole, since residues mod 3 are 0,1,2, and 5 > 3, so at least one residue class must repeat or cover all? Actually, in any 3 consecutive, one divisible by 3; the span 5 covers at least one full residue cycle. Minimal case: the set must cover at least one multiple of 3. In fact, the maximum gap between multiples of 3 is 3, so 5 numbers span more than one cycle, so at least one number is divisible by 3. Similarly for 5: the gap is 5, so one number is divisible by 5. 📰 Bay Area Recycling For Community 8605868 📰 Avi Arads Secret Life Revealed How He Revolutionized Comic Legends Forever 7967713 📰 Unleash Your Inner Queen With The Ultimate Queen Of Hearts Costume Trendsetter Alert 3742746 📰 Zittekist Breakthrough The Hidden Cosmetics Secret Streets Are Trying To Hide 6086587 📰 Batman Batman Game 2428173 📰 This Unbreakable Fate Were Rejecting Itclaim Your Victory Now 2340980 📰 The Medium On Tv Show Got Realthis Hidden Secret Will Change Everything You Thought You Knew 1599458 📰 Tor Network Mac 4817890 📰 1St Class Flights 3473510 📰 The Leg Curl Machine Secret No Gym Guru Wants You To Know 7230669 📰 Unlock Free Shooters Onlineplay Explosive Action Games Instantly 1357298 📰 From Email To Cloud The Ultimate O365 Roadmap For Small Business Success 7635899 📰 Ceasing The Cult Of Dispater Secrets You Need To Know Before Its Too Late 5545606 📰 Bank Interest Amount 7579362 📰 Breaking Wlos 13 News Reveals Ashevilles Biggest Shockheres What You Missed 7444480