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  2. Turboflat4

    Maths problem

    Sorry @Vid, I just noticed a small gap in my proof above. Please use the following instead (I added a case zero just to handle the k=1 case, which otherwise interferes with the reasoning). First restate the problem so it's correct. Among any 6 consecutive natural numbers (integers greater than or equal to one), prove that there exists a prime that divides exactly one of them. Denote the natural number sequence: k, k+1, k+2, k+3, k+4, k+5 Case 0: k = 1. In this case, the sequence is simple 1, 2, 3, 4, 5, 6 and it is trivially obvious that 5 is a prime that satisfies the requirement. (Case 0 proven). Case 1: k is an even number greater than 1. In this case, only need to study the odds k+1, k+3, k+5. Case 1a: Let's say 3 divides k+1. You're guaranteed that 3 will divide neither k+3 nor k+5 (since the difference between each of these integers and (k+1) is not a multiple of 3 -> the fancy way of saying this is "modulo 3"). Then k+3 has a minimum prime factor of 5. If 5 is indeed a factor of k+3, then 5 divides no other number in the sequence (by reasoning modulo 5), so 5 is a prime factor unique to k+3 (done). If 5 is not a factor of k+3, then k+3 must have a minimum prime factor of 7 or greater. There can only be a single multiple of 7 or more in a sequence of six consecutive naturals, so this prime factor will be unique to k+3 (done). (Case 1a proven) Case 1b: Let's say 3 does not divide k+1. Hence k+1 has a minimum prime factor of 5. If 5 is indeed a factor of k+1, then none of the other numbers in the sequence can be a multiple of 5 (reasoning modulo 5). Hence 5 is a unique prime factor of k+1 (done). If 5 is not a factor of k+1, then k+1 has a minimum prime factor of 7 or greater. Note that no other number in a sequence of six consecutive naturals can be a multiple of this prime, so the prime is unique to k+1 (done). (Case 1b proven) Case 2: k is an odd number greater than 1. In this case, only need to study the odds k, k+2, k+4 Case 2a: Let's say 3 divides k. Then 3 divides neither k+2 nor k+4 (reasoning modulo 3). Then k+2 has a minimum prime factor of 5. If 5 is indeed a factor of k+2, it is not a factor of any other number in the sequence, so it is a unique prime factor of k+2 (done). If 5 is not a factor of k+2, then k+2 has a minimum prime factor of 7 or greater. In this case, no other number in the sequence can have this prime as a factor, so it will be a unique prime factor to k+2 (done). (Case 2a proven) Case 2b: Let's say 3 does not divide k. Then k has a minimum prime factor of 5. Let's say 5 is a factor of k. Note that in this sub-case, you *cannot* conclude that 5 is a factor unique to k, since k+5 will also be a multiple of 5. A different argument is needed. First observe that neither k+2 nor k+4 will be a multiple of 5 (reasoning modulo 5). Therefore, k+2 must have a prime factor of 3 or a prime factor greater than or equal to 7. If the latter case holds, then the prime factor will be unique to k+2 (done). If the former case holds, then k+4 cannot be a multiple of 3 (reasoning modulo 3). It must have a prime factor greater than or equal to 7, and this will be unique to k+4 (done). (Case 2b proven). All cases proven. (QED).
  3. These guys work under sun everyday sure fitter than most people
  4. Today
  5. See they also say wanna excercize do it by yourself
  6. Turboflat4

    Maths problem

    LOL "divine" πŸ˜‚ OK, @Vid, let's give it a shot, yah? First restate the problem so it's correct. Among any 6 consecutive natural numbers (integers greater than or equal to one), prove that there exists a prime that divides exactly one of them. Denote the natural number sequence: k, k+1, k+2, k+3, k+4, k+5 Case 1: k is even. In this case, only need to study the odds k+1, k+3, k+5. Case 1a: Let's say 3 divides k+1. You're guaranteed that 3 will divide neither k+3 nor k+5 (since the difference between each of these integers and (k+1) is not a multiple of 3 -> the fancy way of saying this is "modulo 3"). Then k+3 has a minimum prime factor of 5. If 5 is indeed a factor of k+3, then 5 divides no other number in the sequence (by reasoning modulo 5), so 5 is a prime factor unique to k+3 (done). If 5 is not a factor of k+3, then k+3 must have a minimum prime factor of 7 or greater. There can only be a single multiple of 7 or more in a sequence of six consecutive naturals, so this prime factor will be unique to k+3, and we're done (done). (Case 1a proven) Case 1b: Let's say 3 does not divide k+1. Hence k+1 has a minimum prime factor of 5. If 5 is indeed a factor of k+1, then none of the other numbers in the sequence can be a multiple of 5 (reasoning modulo 5). Hence 5 is a unique prime factor of k+1 (done). If 5 is not a factor of k+1, then k+1 has a minimum prime factor of 7 or greater. Note that no other number in a sequence of six consecutive naturals can be a multiple of this prime, so the prime is unique to k+1 (done). (Case 1b proven) Case 2: k is odd. In this case, only need to study the odds k, k+2, k+4 Case 2a: Let's say 3 divides k. Then 3 divides neither k+2 nor k+4 (reasoning modulo 3). Then k+2 has a minimum prime factor of 5. If 5 is indeed a factor of k+2, it is not a factor of any other number in the sequence, so it is a unique prime factor of k+2 (done). If 5 is not a factor of k+2, then k+2 has a minimum prime factor of 7 or greater. In this case, no other number in the sequence can have this prime as a factor, so it will be a unique prime factor to k+2 (done). (Case 2a proven) Case 2b: Let's say 3 does not divide k. Then k has a minimum prime factor of 5. Let's say 5 is a factor of k. Note that in this sub-case, you *cannot* conclude that 5 is a factor unique to k, since k+5 will also be a multiple of 5. A different argument is needed. First observe that neither k+2 nor k+4 will be a multiple of 5 (reasoning modulo 5). Therefore, k+2 must have a prime factor of 3 or a prime factor greater than or equal to 7. If the latter case holds, then the prime factor will be unique to k+2 (done). If the former case holds, then k+4 cannot be a multiple of 3 (reasoning modulo 3). It must have a prime factor greater than or equal to 7, and this will be unique to k+4 (done). (Case 2b proven). All cases proven. (QED). Not that tough after all, eh? 🀣
  7. jorditaylor23

    Road Trip To Thailand

    I loves road trip just because of u can see each and every point.And always wear a hoodies and shirts of drug rug materials.And I am satisfied with their quality and services
  8. Angcheek

    Mercs: property news & updates

    Lifetime also dont know can save up 1m per bedroom or not .... haiz Come with xmm ... then different story 😁
  9. Angcheek

    Mercs: property news & updates

    How this will impact property? Maybe no impact πŸ˜†
  10. Angcheek

    COVID-19: Social Distancing in Singapore

    Training in hospital
  11. Saw the zaobao news at around 7.50pm today. Couldn't believe my eyes. Used the translator and still couldn't believe it. Checked a few news site but no news of 287. Eventually the truth set in. 287 cases. My knees almost went weak. I'm worried and concerned, as everyone should. Worried and concerned for the well being of our healthcare workers, our healthcare capacities and medical supplies. But I remain positive, though bleak. We can't do anything on a national level but the rest of us here, let's continue what we've been doing. Stay home. Stay home so we don't burden our already overwhelmed healthcare system. Take care of ourselves, so we don't take away limited hospital beds and leave it to who need it more. Stay safe everyone. Good luck Singapore. We need all the luck we can get now. All the ones still running around outside, **** ***.
  12. haha, may be a long wait looking at things.
  13. Fcw75

    COVID-19: Social Distancing in Singapore

    They can’t even enforce PMDs, you think they will do a good job enforcing this?
  14. Simple explanation - this Paynpay G is confident that their lightning will sure kill the coronavirus eventually. [Including the oppo?] So do the right thing when you see lightning during the erection.....haha
  15. After what they did, the economy is also in the dumps. Sorry but they are simply always 2 steps behind. Yet always want to hao lian and laugh at other countries.
  16. Aiyah, I just sold a Submariner last week less than $10k before the CB. Wait CB over, i sell you my SD16600 lah, under $10k ok?
  17. err..actually I didn't even try.......cause my english has always been poor. I have given up trying long ago. So thanks for helping, teacher.
  18. Hamburger

    Report Illegal Parking

    So the question here is, was it the merc or the illegal parking got on your nerve?
  19. So many covidiots these days trying to make a name for themselves Two teens charged after one allegedly placed juice he drank back on FairPrice supermarket shelf https://www.straitstimes.com/singapore/courts-crime/two-teens-charged-after-one-allegedly-placed-juice-he-drank-back-on-fairprice
  20. That's why there's so many covidiots out there not taking this seriously. What circuit breaker.. just do a proper lockdown.. Stop worrying bout economy and elections first..
  21. While the number of infected is alarming I believe the situation is still not dire as long as the local populace infected is below 50 every day and goes on a decline. The FW poses a big problem and we may see thousands infected and they will end up depleting most of our critical medical facilities and leave the citizens out in the cold. What happens then..... The govt is doing an ok job imho. Obviously they wanna safeguard our economy and tried to balance this against the risk of community spread. Too early to say if they make the right call. Pointless to have everyone safe but the economy in ruins. Everyone is working towards a common goal. Just be patient and do your part and I really hope we come hard on those who are treating this whole episode with nonchalance.
  22. took 10 grand off the table from US markets but nowhere to spend it😊
  23. From the start, it was a wait and see, step approach. By virtue of that itself, it already means we must be always a step behind the virus.
  24. We ah bang with My mah, we feed and house so many of their JB workers, they only laugh at us secretly, we won't be embarrassed. So I didn't mention My.
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