M/N is the product of all the integers from 1010 through 2018 inclusive. That's 1009 integers being multiplied.

As there are more even numbers and other powers of 2 than powers of 5, the presence of 5 as a factor in the product is the limiting number sought. The number of times 5 appears as a factor in the product is the number of trailing zeros.

The first few powers of 5 are:

                                                                      

5                                                                             

25                                                                            

125                                                                           

625                                                                           

3125                                                                          

15625  

1009/5 = 201 with remainder 4; so far: 202 as a multiple of 5 occurs within the "remainder" no matter which end you consider as being the remainder.

1009/25 = 40 with remainder 9; another 40 as a multiple of 25 does not occur within the "remainder".

1009/125 = 8 with remainder 9; another 8

1009/625 = 1 with remainder 384; add in 2, as a multiple of 625 does occur within the "remainder". The two are 1250 and 1875.

So add 'em up: 202+40+8+2 = 252 trailing zeros.

A check:

bf1$=BigFact$(2018)

bf2$=BigFact$(1009)

ans$=BigDiv$(bf1$,bf2$,5000)

open 2,"trailing zeros.txt","w"

writeln 2,ans$

Writeln 2," "

a$=ans$

do

  if len(a$)>100 then

    Writeln 2,left$(a$,100)

    a$=mid$(a$,101)

  else

    Writeln 2,a$

    a$=""

  Endif

loop until len(a$)=0

close 2

print ans$

print len(ans$)

i=len(ans$)

do

   i=i-1

loop until mid$(ans$,i,1)<>"0"

print i

finds the quotient has 3201 digits; the last non-zero digit occurs at the 2949th position.

2249483470254618793146568878523059738419104216708286338140495265867841665266644649469838759370150595

2666837398682463531426522744442725760046424072366164556550788836187439077072346300761330933211883971

9924782303069407468591493419183199743769051243270110964017112529255653921258626426737409630815954874

3708955879586621838763810511604084110466733602032151839785592755651632613978517754825870999738189562

0457217463264426690478056117483319701483284765165699910637048380928830837961955456341940592180780596

6230203923226319545963135616121710703468809757884919446347361672109620938881614809226726534403961087

8864546519989652455961811857348394281234573586539980083203096556644263215016309556322408780127964303

7898584392037541218379677880318081381229605261786703798085834560838336293907709131282966348733083827

1396658245157835050781437348166298167126156805057425192911753778401549966972731728676045204071868009

8727142355629026599182512807074377148048617039453917609301954698566203568584693858575835464338249500

3433662112755378084397960861916336583723122505731044451225905853927123983160691439309364703138591683

4047482778055145313778905072801563911017255075977071893289002151314010726390443362316115433764200641

3440856930613478237223351807727975074580896447877641992761696968813609155721018221680054590844714574

8961233008415186848715649664848585201249397691645696011322490470258410681425418745294411910429767860

7782713417980282102060172713194655457602182723783680963264605639161175467644372066729831950756024723

7037029752828118623516023165828010472378614853087092437111281359202016800400535502591707175832795349

1184446814238785718340935308006574391596880846666755527766753900430377538489089552579332859842999509

8267113576551839441295127592066876665412703666093063925196150544885654869560845761435669139646376464

1798017764141697881344449346962474562967415067684772728929725797445001493757213490588930139338809838

8285374456858364397677788495980161548612264406640006586531469775983010508617779939750766839892611869

1717441851749948397847806117306474886699598785079444424702579628209222243019344054170952878690284845

1077745597893862506040024773352085874147860797798453357581179069324160671399347231313927261082668266

6586193495438056247502929762819300608114081407856457881435243491478618275599721471108815236830270557

5536985746006462277802197389893742640606513999018845353736190466744125720861699473882442238974388531

6981556851756416668149444878880627761066215945526051465641006671863133980269816880705223987493154336

6323871896356250162254372810597023266298910444384398329581509113813152722949479936459630704123142966

0847922953721469864157196981232459944696782429240125083190958623284012418377005353725725345377396353

7207216841758684837475820284693097231354766971418136378775178626989279159039612771740552308516751689

7682296339537671115130522904955251734102551558954996992547422588789268510680745259980816432568278362

2779992593275834507897886220253369898128673603584000000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

0