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📰 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. 📰 Thus, product always divisible by \( 2^3 = 8 \), \( 3 \), \( 5 \), so by \( \text{lcm}(8,3,5) = 120 \). 📰 375 Pearl Street 5601976 📰 Refinance Mortgage Calculator 9096378 📰 Secrets Of Vivian Nixon Exposedinside The Shocking Revelation 8147946 📰 Indi Sports 2617560 📰 Physical Safeguards Are 246343 📰 Does Masturbation Cause Acne 2704740 📰 Heritage Funeral Home Sioux Falls 1791304 📰 Injustice 2 Characters 9197869 📰 The Secret Peco Login Trick No One Talks About 1089047 📰 You Wont Believe Which Golf Ball Dominates Like Never Before 5533980 📰 The Rise Of Hexempire You Need This Revolutionary Power Now 7753819 📰 Why 4Sherid Is Taking The Internet By Stormclick Now For The Full Story 4422060 📰 The Fidelity Logoin Youve Been Overlookingheres Why Its The Ultimate Symbol Of Trust Excellence 6327929 📰 Finney Actor 5219862 📰 Kettlebell Workouts 8458834