# Project Euler 805: Shifted Multiples - T(200) mod 1e9+7.
# Pure Flow port of the native C solver.
import euler.nt { gcd, mod_pow }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
const MOD: i64 = 1000000007
const M: i64 = 200
function mod_inv(a: i64) -> i64 {
return mod_pow(a, MOD - 2, MOD)
}
# Sieve primes up to 10000 into pp; returns count.
function init_primes(pp: ptr<i64>) -> i64 {
let limit: i64 = 10000
let sieve: ptr<i8> = calloc(limit + 1, 1) as ptr<i8>
let mut pc: i64 = 0
let mut i: i64 = 2
while i <= limit {
if sieve[i] == 0 {
pp[pc] = i
pc = pc + 1
let mut j: i64 = i * i
while j <= limit {
sieve[j] = 1
j = j + i
}
}
i = i + 1
}
free(sieve)
return pc
}
# Factorize n into pp/ee arrays. Returns count.
function factorize(n0: i64, primes: ptr<i64>, nprimes: i64, pp: ptr<i64>, ee: ptr<i64>) -> i64 {
let mut cnt: i64 = 0
let mut x: i64 = n0
let mut i: i64 = 0
while i < nprimes {
let p: i64 = primes[i]
if p * p > x {
i = nprimes
} else {
if x % p == 0 {
let mut e: i64 = 0
while x % p == 0 {
x = x / p
e = e + 1
}
pp[cnt] = p
ee[cnt] = e
cnt = cnt + 1
}
i = i + 1
}
}
if x > 1 {
pp[cnt] = x
ee[cnt] = 1
cnt = cnt + 1
}
return cnt
}
function euler_phi(n: i64, primes: ptr<i64>, nprimes: i64) -> i64 {
if n == 1 {
return 1
}
let pp: ptr<i64> = calloc(64, 8) as ptr<i64>
let ee: ptr<i64> = calloc(64, 8) as ptr<i64>
let c: i64 = factorize(n, primes, nprimes, pp, ee)
let mut phi: i64 = n
let mut i: i64 = 0
while i < c {
phi = (phi / pp[i]) * (pp[i] - 1)
i = i + 1
}
free(pp)
free(ee)
return phi
}
# Multiplicative order of 10 mod m. Returns 0 if gcd(10,m) != 1.
function mult_order_10(m: i64, primes: ptr<i64>, nprimes: i64) -> i64 {
if m == 1 {
return 1
}
if gcd(m, 10) != 1 {
return 0
}
let phi: i64 = euler_phi(m, primes, nprimes)
let pp: ptr<i64> = calloc(64, 8) as ptr<i64>
let ee: ptr<i64> = calloc(64, 8) as ptr<i64>
let c: i64 = factorize(phi, primes, nprimes, pp, ee)
let mut k: i64 = phi
let mut i: i64 = 0
while i < c {
let p: i64 = pp[i]
while k % p == 0 && mod_pow(10, k / p, m) == 1 {
k = k / p
}
i = i + 1
}
free(pp)
free(ee)
return k
}
# Derive bounds on digit length k for leading digit d.
# k_low = -1 means no solution. k_high = -1 means unbounded.
function digit_k_bounds(a: i64, b: i64, d: i64, k_low: ptr<i64>, k_high: ptr<i64>) -> void {
let mut e: i64 = 0
let mut t: i64 = a
while t < b {
t = t * 10
e = e + 1
}
k_low[0] = e + 1
let denom: i64 = a * (d + 1) - 10 * b
if denom <= 0 {
k_high[0] = -1
return
}
let max_pow10: i64 = (b * d - 1) / denom
if max_pow10 <= 0 {
k_low[0] = -1
k_high[0] = -1
return
}
let mut e_high: i64 = 0
let mut pow10: i64 = 1
while pow10 * 10 <= max_pow10 {
pow10 = pow10 * 10
e_high = e_high + 1
}
k_high[0] = e_high + 1
}
# For reduced a/b, find minimal (k,d,D). Returns 1 on success.
function find_N_params(a: i64, b: i64, primes: ptr<i64>, nprimes: i64, out_k: ptr<i64>, out_d: ptr<i64>, out_D: ptr<i64>) -> i32 {
if a == b {
out_k[0] = 1
out_d[0] = 1
out_D[0] = 9 * b
return 1
}
let D: i64 = 10 * b - a
if D <= 0 {
return 0
}
let mut best_k: i64 = 0
let mut best_d: i64 = 0
let mut have: i32 = 0
let kl: ptr<i64> = calloc(1, 8) as ptr<i64>
let kh: ptr<i64> = calloc(1, 8) as ptr<i64>
let mut d: i64 = 1
while d <= 9 {
digit_k_bounds(a, b, d, kl, kh)
if kl[0] != -1 {
let mut klow: i64 = kl[0]
if klow < 2 {
klow = 2
}
let m: i64 = D / gcd(D, d * b)
let ord10: i64 = mult_order_10(m, primes, nprimes)
if ord10 != 0 {
let k: i64 = ((klow + ord10 - 1) / ord10) * ord10
if !(kh[0] != -1 && k > kh[0]) {
if have == 0 || k < best_k || (k == best_k && d < best_d) {
best_k = k
best_d = d
have = 1
}
}
}
}
d = d + 1
}
free(kl)
free(kh)
if have == 0 {
return 0
}
out_k[0] = best_k
out_d[0] = best_d
out_D[0] = D
return 1
}
function N_mod(a0: i64, b0: i64, primes: ptr<i64>, nprimes: i64) -> i64 {
let g: i64 = gcd(a0, b0)
let a: i64 = a0 / g
let b: i64 = b0 / g
if a == b {
return 1 % MOD
}
let kp: ptr<i64> = calloc(1, 8) as ptr<i64>
let dp: ptr<i64> = calloc(1, 8) as ptr<i64>
let Dp: ptr<i64> = calloc(1, 8) as ptr<i64>
if find_N_params(a, b, primes, nprimes, kp, dp, Dp) == 0 {
free(kp)
free(dp)
free(Dp)
return 0
}
let k: i64 = kp[0]
let d: i64 = dp[0]
let D: i64 = Dp[0]
free(kp)
free(dp)
free(Dp)
let invD: i64 = mod_inv(D % MOD)
let ten_k: i64 = mod_pow(10, k, MOD)
let term: i64 = (ten_k - 1 + MOD) % MOD
let mut res: i64 = (d % MOD) * (b % MOD) % MOD
res = res * term % MOD
res = res * invD % MOD
return res
}
function main() -> i32 {
let primes: ptr<i64> = calloc(2000, 8) as ptr<i64>
let nprimes: i64 = init_primes(primes)
let mut total: i64 = 0
let mut u: i64 = 1
while u <= M {
let a: i64 = u * u * u
let mut v: i64 = 1
while v <= M {
if gcd(u, v) == 1 {
let b: i64 = v * v * v
total = (total + N_mod(a, b, primes, nprimes)) % MOD
}
v = v + 1
}
u = u + 1
}
free(primes)
printf("%lld\n", total)
return 0
}
Generated C
#include <stdint.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Flow runtime helpers */
typedef struct flow_temp_node { struct flow_temp_node* next; } flow_temp_node;
static flow_temp_node* flow_temp_head = NULL;
static int flow_temp_atexit_set = 0;
__attribute__((unused)) static void flow_temp_free_all(void) {
while (flow_temp_head) {
flow_temp_node* n = flow_temp_head;
flow_temp_head = n->next;
free(n);
}
}
__attribute__((unused)) static void* flow_temp_alloc(size_t nbytes) {
flow_temp_node* node = (flow_temp_node*)malloc(sizeof(flow_temp_node) + nbytes);
if (!node) return NULL;
node->next = flow_temp_head;
flow_temp_head = node;
if (!flow_temp_atexit_set) {
flow_temp_atexit_set = 1;
atexit(flow_temp_free_all);
}
return (void*)(node + 1);
}
#ifndef FLOW_DIAG
#define FLOW_DIAG(msg) fprintf(stderr, "%s", (msg))
#endif
#ifndef FLOW_LOG
#define FLOW_LOG(fmt, ...) printf(fmt, __VA_ARGS__)
#endif
#ifndef FLOW_LOG_EMPTY
#define FLOW_LOG_EMPTY(fmt) printf(fmt)
#endif
static char* flow_strcat(const char* a, const char* b) {
size_t la = strlen(a ? a : ""), lb = strlen(b ? b : "");
char* r = (char*)flow_temp_alloc(la + lb + 1);
if (!r) return NULL;
if (la) memcpy(r, a, la);
if (lb) memcpy(r + la, b, lb);
r[la + lb] = '\0';
return r;
}
#define __flow_in_arr(arr, val) __extension__ ({ \
int _found = 0; \
size_t _n = sizeof(arr)/sizeof((arr)[0]); \
for (size_t _i = 0; _i < _n; _i++) { \
if ((arr)[_i] == (val)) { _found = 1; break; } \
} _found; })
/* Unified fault handler (MISRA #279) — override with -DFLOW_FAULT_HANDLER=fn */
#ifndef FLOW_FAULT_HANDLER
__attribute__((unused)) static inline void flow_fault_handler(const char* msg) {
fprintf(stderr, "flow: %s\n", msg ? msg : "fault");
abort();
#if defined(__GNUC__) || defined(__clang__)
__builtin_unreachable();
#endif
}
#else
#define flow_fault_handler FLOW_FAULT_HANDLER
#endif
#define flow_div_by_zero_handler() flow_fault_handler("division by zero")
#define flow_shift_ub_handler() flow_fault_handler("invalid shift (amount out of range or left-shift of negative)")
#ifndef FLOW_CHECKED_DIV
#define FLOW_CHECKED_DIV(L, R) (((R) != 0) ? ((L) / (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_MOD
#define FLOW_CHECKED_MOD(L, R) (((R) != 0) ? ((L) % (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHL
#define FLOW_CHECKED_SHL(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull)) && ((L) >= 0)) ? ((L) << (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHR
#define FLOW_CHECKED_SHR(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull))) ? ((L) >> (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#include <math.h>
void* _ui_state = NULL;
static inline float i32_to_f32(int32_t v) { return (float)v; }
/* Host stub for @gpu kernels (device codegen replaces this). */
static inline int32_t gpu_thread_id(void) { return 0; }
int64_t gcd_i64_i64(int64_t a0, int64_t b0);
int64_t lcm_i64_i64(int64_t a, int64_t b);
int64_t isqrt_i64(int64_t n);
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod);
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
bool is_prime_i64(int64_t n);
int64_t mod_inv_i64(int64_t a);
int64_t init_primes_ptr_i64(int64_t* pp);
int64_t factorize_i64_ptr_i64_i64_ptr_i64_ptr_i64(int64_t n0, int64_t* primes, int64_t nprimes, int64_t* pp, int64_t* ee);
int64_t euler_phi_i64_ptr_i64_i64(int64_t n, int64_t* primes, int64_t nprimes);
int64_t mult_order_10_i64_ptr_i64_i64(int64_t m, int64_t* primes, int64_t nprimes);
void digit_k_bounds_i64_i64_i64_ptr_i64_ptr_i64(int64_t a, int64_t b, int64_t d, int64_t* k_low, int64_t* k_high);
int32_t find_N_params_i64_i64_ptr_i64_i64_ptr_i64_ptr_i64_ptr_i64(int64_t a, int64_t b, int64_t* primes, int64_t nprimes, int64_t* out_k, int64_t* out_d, int64_t* out_D);
int64_t N_mod_i64_i64_ptr_i64_i64(int64_t a0, int64_t b0, int64_t* primes, int64_t nprimes);
int32_t main(void);
static const int64_t MOD = 1000000007;
static const int64_t M = 200;
int64_t gcd_i64_i64(int64_t a0, int64_t b0) {
int64_t a = a0;
int64_t b = b0;
while (b != 0) {
int64_t t = FLOW_CHECKED_MOD((a), (b));
a = b;
b = t;
}
return a;
}
int64_t lcm_i64_i64(int64_t a, int64_t b) {
if ((a == 0 || b == 0)) {
return 0;
}
return (FLOW_CHECKED_DIV((a), (gcd_i64_i64(a, b))) * b);
}
int64_t isqrt_i64(int64_t n) {
if (n < 2) {
return n;
}
int64_t x = n;
int64_t y = FLOW_CHECKED_DIV(((x + 1)), (2));
while (y < x) {
x = y;
y = FLOW_CHECKED_DIV(((x + FLOW_CHECKED_DIV((n), (x)))), (2));
}
return x;
}
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod) {
int64_t a = FLOW_CHECKED_MOD((a0), (mod));
int64_t b = FLOW_CHECKED_MOD((b0), (mod));
int64_t result = 0;
while (b > 0) {
if (FLOW_CHECKED_MOD((b), (2)) == 1) {
result = FLOW_CHECKED_MOD(((result + a)), (mod));
}
a = FLOW_CHECKED_MOD(((a * 2)), (mod));
b = FLOW_CHECKED_DIV((b), (2));
}
return result;
}
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
if (mod == 1) {
return 0;
}
int64_t result = 1;
int64_t b = FLOW_CHECKED_MOD((base), (mod));
int64_t e = exp;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
result = mulmod_i64_i64_i64(result, b, mod);
}
b = mulmod_i64_i64_i64(b, b, mod);
e = FLOW_CHECKED_DIV((e), (2));
}
return result;
}
bool is_prime_i64(int64_t n) {
if (n < 2) {
return 0;
}
if (n < 4) {
return 1;
}
if ((FLOW_CHECKED_MOD((n), (2)) == 0 || FLOW_CHECKED_MOD((n), (3)) == 0)) {
return 0;
}
int64_t i = 5;
while ((i * i) <= n) {
if ((FLOW_CHECKED_MOD((n), (i)) == 0 || FLOW_CHECKED_MOD((n), ((i + 2))) == 0)) {
return 0;
}
i = (i + 6);
}
return 1;
}
int64_t mod_inv_i64(int64_t a) {
return mod_pow_i64_i64_i64(a, (MOD - 2), MOD);
}
int64_t init_primes_ptr_i64(int64_t* pp) {
int64_t limit = 10000;
int8_t* sieve = (int8_t*)(((int8_t*)(calloc((limit + 1), 1))));
int64_t pc = 0;
int64_t i = 2;
while (i <= limit) {
if (sieve[i] == 0) {
pp[pc] = i;
pc = (pc + 1);
int64_t j = (i * i);
while (j <= limit) {
sieve[j] = 1;
j = (j + i);
}
}
i = (i + 1);
}
free(sieve);
return pc;
}
int64_t factorize_i64_ptr_i64_i64_ptr_i64_ptr_i64(int64_t n0, int64_t* primes, int64_t nprimes, int64_t* pp, int64_t* ee) {
int64_t cnt = 0;
int64_t x = n0;
int64_t i = 0;
while (i < nprimes) {
int64_t p = primes[i];
if ((p * p) > x) {
i = nprimes;
} else {
if (FLOW_CHECKED_MOD((x), (p)) == 0) {
int64_t e = 0;
while (FLOW_CHECKED_MOD((x), (p)) == 0) {
x = FLOW_CHECKED_DIV((x), (p));
e = (e + 1);
}
pp[cnt] = p;
ee[cnt] = e;
cnt = (cnt + 1);
}
i = (i + 1);
}
}
if (x > 1) {
pp[cnt] = x;
ee[cnt] = 1;
cnt = (cnt + 1);
}
return cnt;
}
int64_t euler_phi_i64_ptr_i64_i64(int64_t n, int64_t* primes, int64_t nprimes) {
if (n == 1) {
return 1;
}
int64_t* pp = (int64_t*)(((int64_t*)(calloc(64, 8))));
int64_t* ee = (int64_t*)(((int64_t*)(calloc(64, 8))));
int64_t c = factorize_i64_ptr_i64_i64_ptr_i64_ptr_i64(n, primes, nprimes, pp, ee);
int64_t phi = n;
int64_t i = 0;
while (i < c) {
phi = (FLOW_CHECKED_DIV((phi), (pp[i])) * (pp[i] - 1));
i = (i + 1);
}
free(pp);
free(ee);
return phi;
}
int64_t mult_order_10_i64_ptr_i64_i64(int64_t m, int64_t* primes, int64_t nprimes) {
if (m == 1) {
return 1;
}
if (gcd_i64_i64(m, 10) != 1) {
return 0;
}
int64_t phi = euler_phi_i64_ptr_i64_i64(m, primes, nprimes);
int64_t* pp = (int64_t*)(((int64_t*)(calloc(64, 8))));
int64_t* ee = (int64_t*)(((int64_t*)(calloc(64, 8))));
int64_t c = factorize_i64_ptr_i64_i64_ptr_i64_ptr_i64(phi, primes, nprimes, pp, ee);
int64_t k = phi;
int64_t i = 0;
while (i < c) {
int64_t p = pp[i];
while ((FLOW_CHECKED_MOD((k), (p)) == 0 && mod_pow_i64_i64_i64(10, FLOW_CHECKED_DIV((k), (p)), m) == 1)) {
k = FLOW_CHECKED_DIV((k), (p));
}
i = (i + 1);
}
free(pp);
free(ee);
return k;
}
void digit_k_bounds_i64_i64_i64_ptr_i64_ptr_i64(int64_t a, int64_t b, int64_t d, int64_t* k_low, int64_t* k_high) {
int64_t e = 0;
int64_t t = a;
while (t < b) {
t = (t * 10);
e = (e + 1);
}
k_low[0] = (e + 1);
int64_t denom = ((a * (d + 1)) - (10 * b));
if (denom <= 0) {
k_high[0] = (-1);
return;
}
int64_t max_pow10 = FLOW_CHECKED_DIV((((b * d) - 1)), (denom));
if (max_pow10 <= 0) {
k_low[0] = (-1);
k_high[0] = (-1);
return;
}
int64_t e_high = 0;
int64_t pow10 = 1;
while ((pow10 * 10) <= max_pow10) {
pow10 = (pow10 * 10);
e_high = (e_high + 1);
}
k_high[0] = (e_high + 1);
}
int32_t find_N_params_i64_i64_ptr_i64_i64_ptr_i64_ptr_i64_ptr_i64(int64_t a, int64_t b, int64_t* primes, int64_t nprimes, int64_t* out_k, int64_t* out_d, int64_t* out_D) {
if (a == b) {
out_k[0] = 1;
out_d[0] = 1;
out_D[0] = (9 * b);
return 1;
}
int64_t D = ((10 * b) - a);
if (D <= 0) {
return 0;
}
int64_t best_k = 0;
int64_t best_d = 0;
int32_t have = 0;
int64_t* kl = (int64_t*)(((int64_t*)(calloc(1, 8))));
int64_t* kh = (int64_t*)(((int64_t*)(calloc(1, 8))));
int64_t d = 1;
while (d <= 9) {
digit_k_bounds_i64_i64_i64_ptr_i64_ptr_i64(a, b, d, kl, kh);
if (kl[0] != (-1)) {
int64_t klow = kl[0];
if (klow < 2) {
klow = 2;
}
int64_t m = FLOW_CHECKED_DIV((D), (gcd_i64_i64(D, (d * b))));
int64_t ord10 = mult_order_10_i64_ptr_i64_i64(m, primes, nprimes);
if (ord10 != 0) {
int64_t k = (FLOW_CHECKED_DIV((((klow + ord10) - 1)), (ord10)) * ord10);
if ((!((kh[0] != (-1) && k > kh[0])))) {
if (((have == 0 || k < best_k) || (k == best_k && d < best_d))) {
best_k = k;
best_d = d;
have = 1;
}
}
}
}
d = (d + 1);
}
free(kl);
free(kh);
if (have == 0) {
return 0;
}
out_k[0] = best_k;
out_d[0] = best_d;
out_D[0] = D;
return 1;
}
int64_t N_mod_i64_i64_ptr_i64_i64(int64_t a0, int64_t b0, int64_t* primes, int64_t nprimes) {
int64_t g = gcd_i64_i64(a0, b0);
int64_t a = FLOW_CHECKED_DIV((a0), (g));
int64_t b = FLOW_CHECKED_DIV((b0), (g));
if (a == b) {
return FLOW_CHECKED_MOD((1), (MOD));
}
int64_t* kp = (int64_t*)(((int64_t*)(calloc(1, 8))));
int64_t* dp = (int64_t*)(((int64_t*)(calloc(1, 8))));
int64_t* Dp = (int64_t*)(((int64_t*)(calloc(1, 8))));
if (find_N_params_i64_i64_ptr_i64_i64_ptr_i64_ptr_i64_ptr_i64(a, b, primes, nprimes, kp, dp, Dp) == 0) {
free(kp);
free(dp);
free(Dp);
return 0;
}
int64_t k = kp[0];
int64_t d = dp[0];
int64_t D = Dp[0];
free(kp);
free(dp);
free(Dp);
int64_t invD = mod_inv_i64(FLOW_CHECKED_MOD((D), (MOD)));
int64_t ten_k = mod_pow_i64_i64_i64(10, k, MOD);
int64_t term = FLOW_CHECKED_MOD((((ten_k - 1) + MOD)), (MOD));
int64_t res = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((d), (MOD)) * FLOW_CHECKED_MOD((b), (MOD)))), (MOD));
res = FLOW_CHECKED_MOD(((res * term)), (MOD));
res = FLOW_CHECKED_MOD(((res * invD)), (MOD));
return res;
}
int32_t main(void) {
int64_t* primes = (int64_t*)(((int64_t*)(calloc(2000, 8))));
int64_t nprimes = init_primes_ptr_i64(primes);
int64_t total = 0;
int64_t u = 1;
while (u <= M) {
int64_t a = ((u * u) * u);
int64_t v = 1;
while (v <= M) {
if (gcd_i64_i64(u, v) == 1) {
int64_t b = ((v * v) * v);
total = FLOW_CHECKED_MOD(((total + N_mod_i64_i64_ptr_i64_i64(a, b, primes, nprimes))), (MOD));
}
v = (v + 1);
}
u = (u + 1);
}
free(primes);
printf("%lld\n", total);
return 0;
}