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Numerator Divisibility Nuance (Posted on 2011-01-22) Difficulty: 3 of 5
Each of p and q is a positive integer, with p and q being relatively prime, such that:

p/q = 1 - 1/2 + 1/3 - 1/4 + ...... - 1/1318 + 1/1319

Prove that p is divisible by 1979.

No Solution Yet Submitted by K Sengupta    
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Solution brute force | Comment 1 of 4

p/q is (reduced so p and q are relatively prime)

 9712435370176457211789032112275705035436058023056668172162691122769643536134963
38888741850351821222770718188739928442132420999158026137218088106604226734001382
25430216309937903018719173066724790037423477319068945022066116849733773498221407
27866748793472314555146081954133359219693224842345716201562838113110488664229990
44325850859533239512961728614358089856690812823243618746637187164109560825785983
26698595871590779728432557411439460051445467200841041264264012542094838584355251
46637078869452056788708944608960016712180409515668301909686056363654039823351264
466923197720153813

/

140044263701948804619343436820447743912828330311279107462977
86781959105584828861694180270900155320736866485178357969757814225700469374818454
18627056015641968611654202902769674287048827771357301459887020639751248870614151
71934261220862316341780802377230654486825981311825976235273131790688782312984726
50458406330723014358986732179004322578669107971003162372262774921109835980551500
76789879745230479962807184918663328472109316343903768300078189869559464609212486
79410595324531407209849932757037376488884192237742369637229672994844934665387927
66661068428697404447683623975519360000

and that numerator is divisible by 1979 per UBASIC, and the quotient is

 4907749050114430122177378530710310780917664488659256276989737808372735490720042
13688095932466812138843212829075254392184144011701882838412373980093090820617171
42713600965102528053925807512240924728359513551828673583661504219168152348772818
23075668920400361068795392599359959181249734634838664073553733255740519789909040
14313214178642364584619367667689787699186868531199403105930867692829490058507318
47750680076599686573235248818312006089664207782132916252786262022281373716197701
59998523936054601712333979084871155488721783484420566907370417566272885206342225
602285597635247.0

10   Sg=1
20   for Den=1 to 1319
30      Tot=Tot+Sg//Den
40      Sg=-Sg
50   next
60   print Tot:print:print num(Tot):print:print num(Tot)@1979
70   print fnPrime(num(Tot)//1979),prmdiv(num(Tot)//1979)
75   print num(Tot)/1979
80   end
90   '

The function fnPrime indicates that the quotient is not probably prime, but its prime factors must be huge, beyond the capability of UBASIC to factor it.


  Posted by Charlie on 2011-01-22 15:51:32
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