Find n where 2ⁿ is the largest power of 2 that divides the product 2011*2012*2013*.....*4022

(In reply to re: analytic solution and computer solution by Ady TZIDON)

 2011 is 27 mod 64 and 4022 is 54 mod 64, so that of the 2012 numbers in the given range, [2012/64] = 31 are divisible by 64, bringing the total to 1980.
 
 Ady says SHOULD BE 32 INSTEAD OF 31
 
 Here are 31; what other is there?
 
 2048  2112  2176  2240  2304  2368  2432  2496  2560  2624  2688  2752  2816
 2880  2944  3008  3072  3136  3200  3264  3328  3392  3456  3520  3584  3648
 3712  3776  3840  3904  3968
 
 Likewise, here are 16 divisible by 128; where is the 17th?
 
  2048  2176  2304  2432  2560  2688  2816  2944  3072  3200  3328  3456  3584
 3712  3840  3968
 
 The eight that are divisible by 256 are here; where it the 9th?
 
  2048  2304  2560  2816  3072  3328  3584  3840
 
 Finally:
 
 
 
 2011 is 475 mod 512 and 4022 is 438 mod 512, so that of the 2012 numbers in the given range, ceil(2012/512) = 4 are divisible by 512, bringing the total to 2008.
 
 Ady says
 SHOULD BE 5 INSTEAD OF 4
 
 I LIST THEM:  1544, 2056, 2568, 3080, 3592
 
but 1544 is not in the range and none of these is divisible by 512. The actual ones:

 2048  2560  3072  3584

 

Posted by Charlie on 2011-12-06 01:47:49