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Title: 1000 divisors Post by fatball on Jan 22nd, 2006, 7:30pm 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.) |
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Title: Re: 1000 divisors Post by Barukh on Jan 23rd, 2006, 4:36am [hide]1969110000[/hide]? |
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Title: Re: 1000 divisors Post by Sir Col on Jan 23rd, 2006, 9:09am There is a slightly better answer... :: [hide]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) [/hide]:: |
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Title: Re: 1000 divisors Post by Barukh on Jan 23rd, 2006, 10:59am Sir Col, I don't know why I skipped [hide]7[/hide]. ??? |
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Title: Re: 1000 divisors Post by fatball on Jan 23rd, 2006, 11:06am Both of you are [hide]WRONG[/hide] although [hide]it can be pointed out that 810,810,000 is the smallest possible number which has exactly 1000 divisors, ignoring the 109 constraint.[/hide] ::) |
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Title: Re: 1000 divisors Post by JohanC on Jan 23rd, 2006, 11:57am on 01/23/06 at 11:06:39, fatball wrote:
In that case [hide]810810000/13*17 = 1060290000[/hide] makes some sense. |
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Title: Re: 1000 divisors Post by fatball on Jan 23rd, 2006, 12:00pm [hide]You got it, JohanC. Well done all![/hide] |
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Title: Re: 1000 divisors Post by SMQ on Jan 23rd, 2006, 12:04pm It would then seem that Sir Col was correct except for [hide]a typo[/hide]... --SMQ |
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