$2017!$

Since there are fewer powers of $5$ than of $2$ in $n!$ for each $n>1$ , the number of zeroes at the end of $n!$ is $\large\lfloor \frac{n}{5}\rfloor+\lfloor \frac{n}{25}\rfloor+\lfloor \frac{n}{125}\rfloor+...$ . For $n=2017$ this gives $502$ zeroes.

We then need to find $\large \frac{2017!}{2^{502}\cdot5^{502}} \pmod{100}$ . We do this by finding the expression’s value $\pmod{4}$ and $\pmod{25}$ and applying the Chinese Remainder Theorem on the results.

Note $\large\frac{2017!}{2^{502}\cdot5^{502}} \pmod{4}$ is obviously $0$ $\large[1]$ , since we have a lot more powers of $2$ in the numerator.

For convenience, let $f(x)$ be $\large\frac{x!}{5^{k}}\pmod{25}$ , where $5^{k}$ is the largest power of $5$ dividing $x!$ , found by the formula above.

Noting that $(5k+1)\cdot(5k+2)\cdot(5k+3)\cdot(5k+4)=250k+24\pmod{25}=-1\pmod{25}$ , we have:

$f(2017)=\frac{2017!}{5^{502}}\pmod{25}=\frac{(1\cdot2\cdot3\cdot4)\cdot1\cdot(6\cdot7\cdot8\cdot9)\cdot2\cdot...\cdot(2011\cdot2012\cdot2013\cdot2014)\cdot403\cdot(2016\cdot2017)}{5^{99}}\pmod{25}=(-1)^{403}\cdot22\cdot f(403)\pmod{25}$

Repeating this gives $\large f(2017)=(-1)^{502}\cdot22\cdot6\cdot1\cdot16\cdot6\pmod{25}=22\pmod{25}$ . So, $\large \frac{2017!}{5^{502}}=22\pmod{25}$ .

On the other hand, since $\large2^{\varphi(25)}=1\pmod{25}$ , we have $\large2^{502}=1^{25}*4=4\pmod{25}$ . By the Extended Euclidean Algorithm, $\frac{2017!}{2^{502}\cdot5^{502}}=18\pmod{25}$ . $\large[2]$

Using the Chinese Remainder Theorem on $\large [1]$ and $\large [2]$ , we get $\large \frac{2017!}{2^{502}\cdot5^{502}} = 68\pmod{100}$ .

another brute-force calculation (C):

void CalcReste()

{

unsigned int r = 1;

for (int i = 1; i <= 2017; i++)

{

    r = (r * i);

    while (r % 10 == 0)

        r /= 10;

    r = r % 1000000;

}

printf("\r\nresult=%d", r % 100);

}

I was worried about the length and complexity of a straightforward factorial solution — some of the comments on that solution seem to express the same concerns— and since you only need the last two non-zero digits, I did something that would keep the numbers fairly small and the computations reasonably simple. I culdn't seem to get my ruby program to format properly, so I rewrote in python (which I've never used before):

1
2
3
4
5
6
7
8
9

y = 1
for x in range(1,2017 + 1):
     y *= x
     if x % 5 == 0:
             while y % 10 == 0: y /= 10
     else:
             y = y % 1000

print y

=> 368

$2017!=4691238606496835154123334763112339349393782860577132281066577713980512516012595872736662937486975615719068118574190055039684498675016335003486162084260814761936137639047147271527784853779168811344440411502275027939415991050284157960514391050429475098827771237860907782270989037431286770001289516610383668883751646289412330746956815078042608777118222417259012153980868059189895322104049920435204499428363782008857646844794331477446552700363905702447931049935526990328994490694834126180948970023199172810067395046387793443712182391229219141687509190169997165999105288234444670380120617845879593015284508182584848776684279192254603481304600761216814359067834700038896121094977112923410962161418760309207304615733033326727689158349196467757174893859766801715038084375406786777201848477381182129928116880468153556380029571594920063448616612270689115465556603807296977350904563171598064124640240795281657800715197562809154611267143627993898729090340625493463471663745464985721404846214479364873455384129755801230728707810565808647033184518995772144586077141286786177949226389412945372413540392151701293281250838958756204163153837443964453600486720525221825699088143204208118159382337325622283474901349749989056015096898103223335186038827380470465855370458598510140857945193760553765566916216709215106775443866275595435116576341611353980221291125608324043415381004668553190615437157153144821431480370310543042783569050552737043116004994954272775368470764353370803652770732775526778496839802549986424954318847883120660842680603299869559738011029563059230656203749758222710387950100134891559949958723716275890640228708151085357244397665962022834876780146100560512504988998836742348623069371030898938525177617865001742687981600631586959407138965695323632627582223327734312982749235859004749859820735685899594668431537644800541514368040344176176871270086881698584165850874718736564614982727941423552997936342261554439093473056598764044455291361446853536071879502080865517129251077344519437553709062740354462849634579031074097030090415955130357923058501875372582622884021698670656041436261732966452366064644993976880037902187704158601140537630618248770944199866376356318213934187140937944723767137643700654528459195195870410092649904500950150741780597441186356193377786517167550891565215014904237613372887196461738112043786909597464188014518671615413369416993155923548619596200244415185235027780256827504921463686375263927146832013739733415790635841012359904207052183025587443484665025845733366318129941870346401189406288528256990010448417262687753689461184135884873596655519352212138152496381261014329391097090581430508962461100475283350105229948713390825455487827973667971985291465417905486215172870422301958038670218271093889911404311340238241609384343471602929057828974212042244985363208139358533532337559620998927693055804700310038772624446222618219985988967483535703342907786830586550412451337390710334238880526922825937662360085910367631733130172048842073097345680732945902792793332602829831845881342607078837770695471847474028645570634513090972739717979821287937915209593755850559390239803377994615251327227174327755382490249111620217682173878162162368797175337613745369408658593635035473721545394209788842820950933730406083012674994776121989133156733827404744726748044985198906318427531927588536843727892922947355179619602102690782250139101505978540203414052107301051746822947826304829786653351633610602523331292516144524374962268687594622651825216895041157061523552903994840389230933302192748488225683327770790601013144883729541833901586537523470811689055307435096846064346921499381593517325591025389942864701907235909325734122252895747126436141382880240115307644605555254795726058174432734944885747513367092411207845289920993484840412169833114859172481675382739585633060075505398416824826637899992393052624550803277726440793041386608456953818518441402496626859430745097801624770601458310257577930955189080930852961325387165055670636172022092056680286777744128449880505172550038920723702055686533489714713321798350249939965601436895513674944952168241082575495811367515595096330432271046353895205411804199350759552430572807491053780317175962789732176752890422249381261747471558765362979518576858038286381500691797637880862587479641943345062089726278051148066707261568289375724636165316963754384143472362440091498928647735569982393814607142507836489799113783892092063008651213950313832972443580359340414193637461478015077548426735742668611239198740585188937133450982506926842126346793965234199388283528961078015173330846502684734302408593410631909164727998098917986980921431211107826500708862552644141524239667145577071378173752771347193464009770531830999814152199488575714185928418720360079792801761757006757097490242923020770556717818013909156071885723319407648776920520010565939892248813275105856499046150143147665052164478648187447721379170746647929936329145435222594106666649552275331070240439673074167099980122644637640284417100774248445182853374733197933146793468983508634721673972146851297139861678861077768481253669544221538764874099003166696286928164699610348065888459891241658772123430238500485897397468489877872689921767403573267503765148377466327454196861314273969113805020243343025595042875536913012078693869653125039075877651453962939257721386318556439989971480675169799103315968 \times 10^{5791}$