Problem:
Score : 600 points

Problem Statement

We have a sequence of k numbers: d_0,d_1,...,d_{k - 1}.

Process the following q queries in order:

The i-th query contains three integers n_i, x_i, and m_i.
Let a_0,a_1,...,a_{n_i - 1} be the following sequence of n_i numbers: \begin{eqnarray} a_j = \begin{cases} x_i & ( j = 0 ) \\ a_{j - 1} + d_{(j - 1)~\textrm{mod}~k} & ( 0 < j \leq n_i - 1 ) \end{cases}\end{eqnarray}
Print the number of j~(0 \leq j < n_i - 1) such that (a_j~\textrm{mod}~m_i) < (a_{j + 1}~\textrm{mod}~m_i).

Here (y~\textrm{mod}~z) denotes the remainder of y divided by z, for two integers y and z~(z > 0).

Constraints

All values in input are integers.

1 \leq k, q \leq 5000

0 \leq d_i \leq 10^9

2 \leq n_i \leq 10^9

0 \leq x_i \leq 10^9

2 \leq m_i \leq 10^9

Input

Input is given from Standard Input in the following format:

k q
d_0 d_1 ... d_{k - 1}
n_1 x_1 m_1
n_2 x_2 m_2
:
n_q x_q m_q

Output

Print q lines.

The i-th line should contain the response to the i-th query.

Sample Input 1

3 1
3 1 4
5 3 2

Sample Output 1

1

For the first query, the sequence {a_j} will be 3,6,7,11,14.

(a_0~\textrm{mod}~2) > (a_1~\textrm{mod}~2)

(a_1~\textrm{mod}~2) < (a_2~\textrm{mod}~2)

(a_2~\textrm{mod}~2) = (a_3~\textrm{mod}~2)

(a_3~\textrm{mod}~2) > (a_4~\textrm{mod}~2)

Thus, the response to this query should be 1.

Sample Input 2

7 3
27 18 28 18 28 46 1000000000
1000000000 1 7
1000000000 2 10
1000000000 3 12

Sample Output 2

224489796
214285714
559523809

Solution in Rust (1.15.1) (judge verdict: Accepted)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use contest::scanner::Scanner;
use contest::misc::{arr, ceil_div};

fn main() {
    let mut scan = Scanner::new();
    let k = scan.read::<i64>();
    let q = scan.read::<usize>();
    let d = scan.readn::<i64>(k as usize);
    let nxm = arr(q, |_| (scan.read::<i64>(), scan.read::<i64>(), scan.read::<i64>()));
    for &(n, x, m) in &nxm {
        let mut last = x;
        let mut eq = 0;
        for i in 0..k {
            let cnt = ceil_div(n-1-i, k);
            last += (d[i as usize] % m) * cnt;
            if d[i as usize]%m == 0 {eq += cnt;}
        }
        println!("{}", n - 1 - (last / m - x / m) - eq);
    }
}
*/

fn main() {
    let exe = "/tmp/bin2170B3D7";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &[u8]) -> Vec<u8> {
    let mut tbl = [0; 256];
    for i in 0..85 { tbl[TBL[i] as usize] = i as u32; }
    let mut v_vec = v.to_vec();
    let padding = (5 - v.len() % 5) % 5;
    v_vec.extend(vec![b'~'; padding]);
    let mut ret = vec![0u8;v_vec.len() / 5 * 4];
    for (i, ck) in v_vec.chunks(5).enumerate() {
        let mut acc = 0;
        for &c in ck {
            acc = acc * 85 + tbl[c as usize];
        }
        ret[i * 4..(i + 1) * 4].copy_from_slice(&unsafe{std::mem::transmute::<u32, [u8; 4]>(acc.to_be())});
    }
    let l = ret.len();
    ret.resize(l - padding, 0u8);
    ret
}

const TBL: &'static [u8] = b"0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz!#$%&()*+-;<=>?@^_`{|}~";
const BIN: &'static [u8] = b"e??420s#R700000000000suY$0RR91c&<SJ00000KmY&$00000000000000000000Kma%Z0sue&000000RR911poj50000000000002M$00000002M$00000WVQhS00000WVQhS00000001BW000000RR911^@s6sGI`;00000sGMT}00000sGMT}0000000000000000000000000001BW00000l9y%WRZv(V_y-LZ00000@Pz{a@Pz{a$N>NVsQ>@~0ssI2`}-mPe??420s#R40suY$4PXO64;%7r76kzCfdc>&fgQVhH~<F)5C9J=1pojoAy3#q2k2x20ByGW&L99c1_R(`7YAXVoseUA8ZF2*05{o!4K@S=$g&4OK_=u0Bme*eU`JT62Mcd7C;5F#*vM@FP~~)FH_RI$Pslb0L)-}f1-3gZZ&5b}004vqJ9!WQQg5d)goJ${82|^l00000002k;{~%)n05zro00K!30RP`~%LD)v0|Q4+RRHhYeETI<M$8Pq{r~?}LG}D9*{*R49yL%{!vH~7K~^!qK^0U<gUtW-_W%D?Q%Q;bOo`7<jVuC4gX~C1i8LewOe6>sbN}t`;Y<f8@~#e4Bp67Glq7&z6aSI7ZTAnz4U8m+L?j?c_s~j>e|!7e93Vj)D@~1i0TWy#5Cs0{4FQ;RYrp3HNQJ<3eoTu%1S*NgI$%B5?ph02Ndd(NDp}jMZ|zGP4~fz9qy)hdbbz+DZC?q_0`C{u3K6t+-)+Z!NR2r>MMA~|7yw~sv~7bu+e3sti;ST9AMWk{_T8pIiNu1m)x=0diQrpZL0mxt`PlCNUO`{ORZNWy=J5hd4X*|j&<q3C@O!s4G7UkvZ~=_vQw|6eZ(E`J1=M6CTo^JF-%tW`3Eg+QX9scA1XlJW7>Rr&3=CI)cGteW1imN+x8REusy#?C?Y-UJZ87Ivll;H`F68hP00FAKd%K-nR|AVTjRl^zhKmh}d+=wty=p}<$rnAx&0k7pklVJ~zO7SGfIoK#$Ov1&!%h2d@7r(mcL+&~J_1Gv{}2TmcnARdu<zY;2^0Sn8+@>PFaIS95O;Wdzh*8EAA|q^RRI3rT<A8#Nc}5bhHi+BeA_SFJ@)pW8dQp{2s{E0gbM@q`-Ka{i7c{>Y$OUyA?q?raQkjs2S92AM2I9XOhXWDz#i87|3DA6K|~F0`;XmZB%MtTTRuS#5JfvFAp751<S;w|<RHX6Aa@i<xd1*^+wcBi0nZPk!4HBPctwQdcq(OcA7U?TpgRP0kh&d%00q&O-M$1C=?DUJlN!4fHoQN<=?2wS(N$M85t@D2(0%>}4}(g>@deUq4DsLp8~<N&M-R8?C+tDN>;{A7Kf&b)Pr9jLo3H|40|HAMQ6Bi-@qdX^5J8O;EP=wnXl98N2zh`$Zr@n%>k^9uB*wO|0c=~O9qlk^U2pU@NF*^fJY5MsDBY>pd^`72!-8U{jT<1oD{*dG#FS^Z-46>M@@*Ho1`<1zjw4|qZ@=~rO%t{Vx(D96y?}zSaD@4J1fa5Hx5GC4-UKa0(045Y7wa8U&kDl~vpFm}bXTOI84kU#o3K>EZxLGHupI$(O+kR9f8nk4buEqf4Fz()w{*LNz<C|*(=Z5lLUJi{SoGocKf+A)$a0!N#s&fF1-Nbz03$$-ya0vQUSSYnguKpd9>az2wj5ARM+lNB>N(N={rK_i1=9}mNCDGT1@8Eb45eLdyIVQgphp9zAx%Bnr3r5b+jKDzNIH09j&H-?Cd5sJ<aKiD1>YS*8L{+9ZW^z5SL<j<t|Xh=cAt9Zz6b{~(jkRBRCPAj+@(yaBqSy`RTY17a)52?B`;MRYml}d6HSRGVP3^txbfP{7{2-&g!2X?x!jN*ME@_<AM?y@T?7zV%&G|jcz;W`b%T2-C<+iD@x_Y<@SCizZxm<<0d+_|GuAci!U)!If5UHvwWWtp{}%#T6%>IF!pIMgNP+D5z<12?9tE2N#7oY<4Fvc<em9D?c5nFqwF#OJSHKSuGD!Cp*T7>3u%O!_gA9;>NpJ)PW}oL^P(a5zL%cNh1Hg+~0dF_~HGyf|E)-~0Vcv~g4!;Wm_6s9T-hDy;*1rb~l4Hs7e(d$fl_R;iT04z&4K~}U#C5*zrET^X7$sG{03E*|x(vAACv~|Z?zsN|#qN8U*(s{i3LSzFP&>I;gc$i*J-JyoS8!ccZ7ll?WIb6l-UzpO3%>*xMnf0ukNqPSw8sE#VQFs54=4qLumb(k!j5l1LpTi9)c41!t=S&7{OA`c2he1JZMCuPJS8ZS14A<pt0iqmbB}_JfFKqQV!;7p-PVKd$q$Yn#YsDb3gpFRBYdPd=O{~U0U5HbpKVZ8L)Rj0lb9(;)Q_!U8GD3HgF66N{sSc3Zr#FFV|#mUbQCW`AYG$vbcQH}2|=37+gjHSzhpxoNi}~2L4CY!t(7%=93}&<jU9A2gME6Ti}Q2{%DRc|UE@n#2Ba|lW3-9c_+#u~1pxW@19f7F*oegLup81-YSjc`nL-bH_6RfUA`gkf5{d98-I{&dFO~{XP?voe(Yxf6W1Qh_2);(fs|R7jga|M=mjPFHd#vsh>xeXV(G_HJG`>eKG&@3h-el;$0T4Rm*zVRUFwI}lXc(k9cEbF?)=A=VCYfYxw;8{$IJgi?zOxBBg84WnG%y%Q(O-2Cez>w20n!~Xc4CKf!qC9`Oa({}G96StEYwE96)Ym~nGcU++%41%P3aPizX-yEZ^L$yautz97^2v3Pj<S*hQ}K|RDZ_52`D0U49Hp%%DNy%B=^(@9?lz<1C9o4?7t{hEG?ouxP!)6Ge9d~$3P(AHXh8jw7TF6sM&VZOJH?li4LfV8*|0jEUT>t*q%)>82j1?YI_Rr0!>$Lv#%Biu5m*+J2`e29g=v)d%sACqj7gOIWDs~nh32q7=QD9Xb{Fg7&v%)i9;EQOB}=Y+wAwpLnz%3D4r}sFvdebDY0ymv!07DK|}+>32*=QbSYboRxI3+9+Gj11+f4<f(Aeig4rLuSOIaH)nZ#GoN@;!iYD+5TL<tij(3=fEu9~ntR{hAe;IZq7etQ_U;zLof~g0#3uqQ#YBghvP=Pir3vBjW*G$_LKn2YNNRVojIQA1o1LTh|3<!PO5{C*Che*2W$^)g>PM{Mo+XdHOH8fQ-Xa{gXwwzyhsRaR411oh51;+n$NegUSY<E{o1Cjb=K_r6(+_?%tZsc$Oj($tg&gc{SQydC|!_b5K338eE+jkb3;7Rx4a1OKrg^q9K@P5Q|)J?+%AM8`X)ah_{W$zVp!>wJ`#Gwf32q<=|yWQSvxe*@tNJtNWDkCO#3Ae%)KcR15Dq(03z1qeUsgjFJuhfzn(EDJ|-gmSegg8<J?ux}U!~b>m-o3f#SOUBxxFqWpLB4A~5ElC?c!U<2UpErl<QKFWNFP{)4j`v&?8mtnFW{>7ZfqOpET?3h5Dkp@f}u!R7Ga5`B$yKa0Kp{ajU>o1+wYfsNjON0&%(tD#sJOLAx$s?Sl~sV!@b@^#Do2_4MK!d0oZlANQ-PK8$^JG+3K{xf^{2m*sd@*6At}MHV+R1_P7X-uH+8v^KEae1W8E)#u$*e003-o-|c`EPyW)#4E0O_ZDV{dw}Zg`C%`f7&474eTj(hD4gy9;i&hXv;`U_aVi8h<`n3{+@8A1v64D41)V)edjVuTYuZ0bGMDhp~Ore7YZ3*E!*~K)z5qn8F^Ea`H=t!A*e)NP(5jk|66y=qRgx;nS6S6kh#sMOwZD|9y&<d{`0ckHdwr_Pw4@J!h%O5;B_jR$icd)6567-v2OgZOFZ4YbkCQUiuoUar>L--1xZG%g}m%s}5(Me^21=lw3$pRA)5Yt1GI2(iqiP<I^`Q&UQY!29N&jP52#Y4d&g9cB)K;lyqy#X!~ZTUCw`*WzsgM9!1ixDK=LsUq^+_@ehH&PN-BXYAr=vPMw0tFB|WEZPoh6WWV0mJS#YkW;SiQ-I9)GOZIcWi9{coBddU=>BQ=?=TYLz4fF+btyz<Sk)Pa2qn;URs6i#YF&pw{^%^djW0{RA7rm;|*|Cp=6K=<P3~V?`^xKoc~Y-gH{2GM6dz6zilBo6UvJTi&6tEU_dz#MtEa91T8cYILLDqvj0?gI9-IZSf4l!gm{zsjo&;2ZxmV>#u!XHv@tELNDmM+V2?GhJ-DMlh@q!oaAVu)wEanWiFdmfJ|j(Z-R>RUw&_kg01Ob0Y*y*v3B4rCDG-B)p(iqJNiIAK3@0s3id7^(9}e-6PRSu3_=}Vr9cio^<0SH^IU)hWNidAG2FDD%<KY*NMF;S?^%c@n)@v7MKp7K6@PoWLh_!7M?+#MDH#>nx7XbZg3TFToB)$*cX$0gLi;ga~ozaR7Q@jpO-pFO&1_U(lP3W&Q&Vb2{P#6KyJ^jO<1LnC5#XUcAH9bEt67np#26rx)y}-$W9j#M)Y{8f&zSsex48Wa8Uf>YB0$6|<M1~1$igr~9Hi_nLd%~^v0wn2Z(DqG;4K)zt8QHK6I~TFAnje%Lxdj=guAow$3dzZhp2m0%MzDThNR1T2+~DrISI!GS9SKMkb`JqtVeCdw3!-M^Lv0jG#^D<slD=`+jW6eU;H1TlSO9e#!>(-)2_snMq_9z|+f9OvuUP#OMXk3HTGAYTQ8@uT2?kPKG`<Kq=>`%?Ktw(D&<Wu%5;AK~Yz3*?BU3%R$tJ!dRUkCSo*$aQIb&TbGVEo)E&;!0z6gH5C=iHUNRCKQ@mmChn8E-6{swBn0v^}Y!uUeh#EouXMgx9JIrw!4+Yk2zSP%koK?_V4(il?@M#P8+$tnRr^hj*b0AdKJZ>1#`da22EI`cGt3ji*j^hw7C0p%AWXEX?Osd@If0sY>aj3o?W&mi#WO4KfJ<7T&h$As2Hez)JYH&i}Afd73s4{*RvW=R>uNn=Iq_lrCvNp1fS2Cxcd>t1*kT4`q^TIUF07=qOWnn`E#p__YC_c{J@RR}Zz#_1LRZ<`Lsk(I=B2Z{zkhzIKo+^y&KbmBbwWkc@);OJ;=AsPk9*}<<71B>yti)PbtxR|oq0Q^C;|5ipPxN%yziTkDSr-3{JC?JX-8Q_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