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wu :: forums    riddles    medium (Moderators: Eigenray, Icarus, william wu, ThudnBlunder, Grimbal, towr, SMQ)    1000 divisors « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: 1000 divisors  (Read 1466 times) fatball Senior Riddler Can anyone help me think outside the box please?     Gender: Posts: 315 1000 divisors   « on: Jan 22nd, 2006, 7:30pm » Quote Modify Find the smallest natural number greater than 1 billion (109) that has exactly 1000 positive divisors.  (The term divisor includes 1 and the number itself.  So, for example, 9 has three positive divisors.) IP Logged Barukh Uberpuzzler     Gender: Posts: 2276 Re: 1000 divisors   « Reply #1 on: Jan 23rd, 2006, 4:36am » Quote Modify 1969110000? IP Logged Sir Col Uberpuzzler impudens simia et macrologus profundus fabulae     Gender: Posts: 1825 Re: 1000 divisors   « Reply #2 on: Jan 23rd, 2006, 9:09am » Quote Modify There is a slightly better answer...   :: As 1000 = 23*53 we are looking to distribute these factors (minus 1) in descending size across the primes. Clearly we will use 24*34*54 = 810000 will be used. So it is the best way to distribute 8 across the next primes. 4*2: 810000*73*11 = 3056130000 (a candidate) 2*2*2: 810000*7*11*13 = 810810000 < 109   But 810000*7*11*17 = 1070290000, which is the least such number greater than 1 billion which has exactly 1000 divisors.   (Barukh, you used 81000*11*13*17 = 1969110000) :: IP Logged mathschallenge.net / projecteuler.net Barukh Uberpuzzler     Gender: Posts: 2276 Re: 1000 divisors   « Reply #3 on: Jan 23rd, 2006, 10:59am » Quote Modify Sir Col, I don't know why I skipped 7.   IP Logged fatball Senior Riddler Can anyone help me think outside the box please?     Gender: Posts: 315 Re: 1000 divisors   « Reply #4 on: Jan 23rd, 2006, 11:06am » Quote Modify Both of you are WRONG although it can be pointed out that 810,810,000 is the smallest possible number which has exactly 1000 divisors, ignoring the 109 constraint.   IP Logged JohanC Senior Riddler     Posts: 460 Re: 1000 divisors   « Reply #5 on: Jan 23rd, 2006, 11:57am » Quote Modify on Jan 23rd, 2006, 11:06am, fatball wrote: ...  it can be pointed out that 810,810,000 is the smallest possible number which has exactly 1000 divisors....   In that case 810810000/13*17 = 1060290000 makes some sense. IP Logged fatball Senior Riddler Can anyone help me think outside the box please?     Gender: Posts: 315 Re: 1000 divisors   « Reply #6 on: Jan 23rd, 2006, 12:00pm » Quote Modify You got it, JohanC.  Well done all! IP Logged SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: 1000 divisors   « Reply #7 on: Jan 23rd, 2006, 12:04pm » Quote Modify It would then seem that Sir Col was correct except for a typo...   --SMQ IP Logged --SMQ Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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