Problem:
Score : 500 points

Problem Statement

For integers b (b \geq 2) and n (n \geq 1), let the function f(b,n) be defined as follows:

f(b,n) = n, when n < b

f(b,n) = f(b,\,{\rm floor}(n / b)) + (n \ {\rm mod} \ b), when n \geq b

Here, {\rm floor}(n / b) denotes the largest integer not exceeding n / b,
and n \ {\rm mod} \ b denotes the remainder of n divided by b.

Less formally, f(b,n) is equal to the sum of the digits of n written in base b.
For example, the following hold:

f(10,\,87654)=8+7+6+5+4=30

f(100,\,87654)=8+76+54=138

You are given integers n and s.
Determine if there exists an integer b (b \geq 2) such that f(b,n)=s.
If the answer is positive, also find the smallest such b.

Constraints

1 \leq n \leq 10^{11}

1 \leq s \leq 10^{11}

n,\,s are integers.

Input

The input is given from Standard Input in the following format:

n
s

Output

If there exists an integer b (b \geq 2) such that f(b,n)=s, print the smallest such b.
If such b does not exist, print -1 instead.

Sample Input 1

87654
30

Sample Output 1

10

Sample Input 2

87654
138

Sample Output 2

100

Sample Input 3

87654
45678

Sample Output 3

-1

Sample Input 4

31415926535
1

Sample Output 4

31415926535

Sample Input 5

1
31415926535

Sample Output 5

-1

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 num_integer::sqrt;
use contest::scanner::Scanner;
use std::cmp::min;

fn main() {
    let mut scan = Scanner::new();
    let n = scan.read::<i64>();
    let s = scan.read::<i64>();
    if n < s {
        println!("{}", -1);
        return;
    }
    if n == s {
        println!("{}", n+1);
        return;
    }
    let sqrt_n = sqrt(n);
    for b in 2..=sqrt_n {
        if calc(n, b) == s {
            println!("{}", b);
            return;
        }
    }
    let mut ret = 1i64<<60;
    for p in 1..=sqrt_n {
        let b = (n - s) / p + 1;
        if b != 1 && calc(n, b) == s {
            ret = min(ret, b);
        }
    }
    println!("{}", if ret == 1i64<<60 {-1} else {ret});
}

fn calc(mut n: i64, b: i64) -> i64 {
    let mut ret = 0;
    while n > 0 {
        ret += n % b;
        n /= b;
    }
    ret
}
*/

fn main() {
    let exe = "/tmp/bin08E7DD92";
    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#R400000000000suY$0RR91IIKYc00000KmY&$00000000000000000000Kma%Z0{}n(000000RR911poj50000000000002M$00000002M$00000f3*Ps00000f3*Ps00000001BW000000RR911^@s600000000000Kh>2000000Kh>2000000000000000sN5j{0000001yBG00000QRQ@G1^@s6000000000000000000000000000000000000000000000000005C8xG00000+&hJ7RZv(VR0$0h00000poIefpoIefkO2SymH+?%0ssI2`}-mPe??420s#R40suY$4G;rB4;%7r76ky{fdc>&fgQVhH~<C(4*(7;1pojoAy3#q2RL5?0Nd>Q&L99c1`THy2V=rMJ0Un4Er>M$H^Bz7ZU+MdH!v;o`<AfCH2_fMbYwTo7KFTzZ3jaM019j~*aSCG1^@sE3wH7l08($KU<e8O;1~c0fdBvi0001x0RO060|4Kn0RRF?4FLcD2l7x@!vH~7K~^!qK~+IiNrTM)RZ~f~xBvf%{!EF_PK_)ANQ3N1NQpEg156|c6X8so`)_v#DDtijR3sQki<BgQJrno-kGyRU$PJ7nh(shHNcYf6jT|8V+uPqk94k$YdjS(%BoGAt=nVmryhw$>TKhHkbbd^WKm;m@$2wr|xpvnJSV;lJ2P#=h+O~ab8xM)m^P~j95_EvJwQc(e&I0cj*$NS~ezyB=J4lT=JVipr1Q-BeV*EpH8|>MHK8uW?`XBD5Z~wRNL5akI{ME!rM2X;AT|rzy171O2n}2uz!&OX;4L0opOb!3+1{EL(12uwg4Zn8{LAY=MjO9}f2owcmW%jlex+7c|G85lW0&Y<U+kLltZ5#wrJ|q~4d?XAEQw0R}XLs!eC<e1Aixj1WNHOg(+1u^i=UwCczyB`e@D%_7l3d%nx7$|(i#Lr0{;`IO4dHugMHk+=+cC)(J%`CXQIckmty55M+qV0FKLT?I$Ov1&!%h42cL+({Z|~cSJ_1Gv{}2TmcnAP=2^0Um`>^j7b9}ISFaITC5N0k9AB0Qxh7tcz6<o{@3lDCO-Hx!>y9S|TD83PT05m)=G-iAvL@z51f$aEj;qVB61)BpO%`m?9&CT5?0KW|cbWVGN`2P=t59D(T{{=!1SHKSum%thau#rIq8K97Wv(NKwNpJ*UP(a5z3u8aD1Ok97S^;l3C^c^^6xtzas3As;I5@ux7qI^rHQq7OLI2jj2Mm7gIX2oUFj~iR<2iEMXd6%vHWl`%Tzb2&``h*x7y@)vz5pG+AiC24xw9GEry?G>{{V$4LiW0EGSkI^5KtH|S-Q?`Zas;7SylxsOc8E(81(u*>3s{o1Q+&?mE7KWLj*!x#{g|%X>QBI7b`yxC<PO+?n5|o?O}Uq&&-R*4-ow57byq!>)x8sWPv;-D3Sw11ltb{%{}&u1SNuvfFKqQ=)t-_nzsRC$q$Zg2?7s=3XF@*+WVw9Dr`$_p93;vTWwHP>zH`$+a@VUs34E69G0<1O@liCSf&Fc+d{)FRbzW-Nfa+bzuoQIAn*$)0d)yMdmzlzTtgs9OIz0tHGc#_eFXW5(&KGw93}&<jU5E0N`>>Cg(8axV&gaN*xrL;2Ba{kUi^vgiv`cdFT9W|i34?FiP(t5TX)!<OwP<|&HQ1R)`LBJ1KaB&4~fGP1TpaD-I#sbFO~{XP?ztQ&>3&NBmIGLX`BeYM)cTlx#d&5#;e3IIF7XYL6-qiT<eH5_SPf>cO9WTYez5FT<E?5a%@+g5CRM;ugqU@Z9MCi92g!zNrHAVwp-FB8E>yRSkK9XOc1)6C+W7NCEN!9Rj>!)5iXF>9m0BGA$Sc9yk8jw+7Fl>QUPomwh#Xc)D<YcMF0+ul4IPx2c78?^uGwf9CwCq!*&b!Mi`-abWq0{x6=h$yk^F~2q^P*EmFrN!^yfI2kb#|PaS7Ol`y|3SH?2DY{V`M39vIjxfz=_D-m2kAhs=dbkOlKU`Yrw3y^Ez*gBLWgKUo*MPTe}Z$6y}V;c(JbFSHD{R>YPmt{T1ZknbL6DTXcNPt7x<X}J=eUhN8{LS~>Iam<JKo~eED~UxJiA@~#eQ#_NeY?j+DAs11U<H?@`Th|_FvdkdC?F$2L`4n=gYbm*bSh4cNz!#Y#%)3<0tHX<Js<|;AUKErkV~g^r>Ct2C+%R2Xwk#w3bSy3Eqx7m3d0OLx*YEc4QzD^-3ESxO(jVw!+j?}N^vkqx^3j{4{pE<7PbP&J!}JiIF^3cv=HnYbq*%S_BCw%ZSe>X1x<zah!`sd-_E{L!b!OR0DvR^1al<Nwhi{X9Y7D2#Ywmr5N*V++xG6q?K}y?KML|@Nn^)fVtzOF?gd?|a1UGsWd|s870B`rSjaZ^?dHG+QNaTc4a%$<&<pMM{q|hZ@|!HeOfkp@=oBu&2Nc*I-b3hG36G!n;I_pVTW|G80U6N(u)@IjIq!BU{);0+yOnMCAox0o{Ua2`A-sLZa_8$G2Keh5R*!{$2a9k|0q83^#dqPhTcdmT{Q`V^;C@EA0#cA#7wKCN+HUQf#w5-eH2?3m?PYUFnLzm)*H?e@OgX*3xv|$=1v4!1e8TH7xJ8(450@^G$OvE3Cn0xho9#L1T;>762wL~Ul(2l?RgO?iIm<CjI~VH;cWN1;e^u{=ZyWhR?(vz}jsEA|&0Y9V0ZBR3f=<+E21V4IOcvqIZ3f~gV$OWbqC4MX+uwj?M2O2e^BLX`jTkZLO6I!T4M9>*EZB<=XMJ;n$1kY^$z~t+VA~fV|3T6J(RHqRGTsZ@{A^Eer9r!^J{#M&-L>luZ}v<9-XrT-^U~J2!hKj>32Vlv+le6G+crg7$Wlr7`*hQ=%eIG(9T5L~+jfw*c4X7*0uXxBb))y-LgAV&I>*4J{tsRuLd0_w1dq5oi%kbcYQz`|!doLCF%q-TFaq9w-J8Tf4lWx|5?BHF;2A`kPz>N;+cX9<^<H3j6ig3kYnNIHqjU=$2WZKcZ17;^W3rH4YL39B!|~R{$>(Mn&5_*_aT)xwO<OsTL98VhX1B-S>*;hhE5$uy&Nsdz%N+^^F1!JFIkJ}>z6dnGC=f_%o^>HfPJv5nDw|l?vk9OX$y&9AV306>elYdM4a18^ZI&tVP0<g7t8T;AlRFOcu@8a^R(21UZktIAO*z@&N7?Klg~D5-O}z)(bTQx9{IT5{i`WuzW=ZJeb#mD;_;#@kM;$}Shj1DZS{OK2>lRt9+e)!gXUfwdT|6~+p_^27Op_!eCN~^lo6S`fSL-D&Eo~czoE#IZCShJ8%pyodYksgVLZtpG1|0o5F-W{b0Do119|a1O<DP~&2!R+ecsb#IWl`I;hWK?r3~V87jPB9P#%}~jULXX=4j!X)5fdDWSv&&wbB|u=4r^~@BqGWmGnco*n+YvL!H)(#0mADW#~vU^&!OHt54M9Fv|~MVYX~GENufkx>8lRGZf_6(|NkS6SUk@WDveF|fcXV-!$5%jAauPz?JvM@FU3HF?Rl@lNWtm`LJx1-hubtN&He_)5nV_WVh^LjufTKPa+J0KI2WhCtpKNw0rm(P*lJ$jjY?jZbS8?HwE?nv18?6b`zv$@1TA$wbc6W;bocEBH|o-KjEz?`0r~4MBjn134+aK{v2TCrY1lr$5paXpgLqv&Sf#B6Efo6kf1CSuBVBY-pfZET&_SdLU8QdScezMw+X`^AHNpY#+i~64?eN^|rn-PK4<9cg&=+uS<a01Bi-jpKFnY(<tu5Yb4yZBhL2rb^cMh!&vl)hFnYB4!6B-b;rr<jhVi9&)brjQN)XVlSVj8wUh%7K!#)<B*LU<9}(i*S@VwY^|DmW)JI5rSs5!ci;w?)EsL=AM=5w^xL+5+u7Aa|gJKAgaI(M}7x!Gp=DcqL_rb-=}ky5I=KWQ<R{{SU9m!RZJoOAmGq3{4NW=rG<wcI^L&>%r>=gXKTL<p@5y0#aY_7zczIjZp$xjo)v$_mmyLfFt|7^Ja+@2y?-h%@`qkb1Q5g1mJ+lz=%^Iq<G96g<Av7fOPNJma-)ymjZsa1ir(z1R!*XAZGBSc0Jt*Zdmv>#+U7cd~#q_P`)dOU>AX7gTT555R4MUz$&Z*(j2Z~E0Du4LxT2)Q2!S(z*!ZK00S#z4^GF1FtAS7M6!);Ul-sTT}A_bH97co2Z#^?)omZ#azP6j5~>(e=u6<S$C_*q0Xgjc^AB(S#!DT|`*k|=9)Cs%Nyi2PxHcU4022D7sUebeqq?~POH!jZ$&!H}=}JTvo^Ggq;MPMRv_5tJ3mbadeK-$rz)t#p#HKur1+m{GNp1fS2A~St=3wa--_coA=LpEeZY?mX*2hKbHuvp9WQ%i}IsS502s8oK&CsP>G339ImBc#SHUs~P20@4i>kP;O;yhz&(BQWCLl{lq=mp4cf#z^YH4(aI(+3PJ0^+zVAPf*-I;<~XFr(|kI9iJ?$p^H76G!^(2TXbqyxPgaf%jJw6dFt&k$FdSb%BQnQa85D6qbDhp>;RwY+&yxb7LQ`)wAxlM`3~OVK#I(75frFq6-CX{1}Qr5G!>(D^tZ8TtjORAm%&jOo5v_;RQ>l6Ikn@Z`bwPt`t@6{4sRS>j{ul5eiSxibcb^4)iE(XoA8)#^K2!+eQJp+xzxoFzH4R>l%t9bPhEKhVHiEByETu>jVh`Ibs<BfjD%JfQC;rkGzW?xe$%hMOepa!wLF^g4qs+DCg7=>G4Fz*3bqbXzq0gKE;UGVUx})&_oAZNWWFNm{Q-djcD^m2dxf^ZcvGWwvWI4_QeEr9S4KS2#HHje#$`lTXO*xF$%XBp92~Mnc`hAl!PQdNoAl_XhMHO61{`lFA*?}a92Dmw!+=k4?@rgw^ga>Y0tm|05@<{J#(ma)ZOh@ybuf|zeXv~hW~*Hb3?4!rY*+HyI>B@n`;jtgf@J6S$2sR&O)tgEa8|ZzJPd3if0`=EI-m%6<i#94Gw^893(gvlztYja&|75#%{EQ_6(tPI+kc?L{~51xaYK9a@3iIQDIs=0k{LUw*&29bWtNp_H;)HR9I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