Coordinated Clockworkers
The Society of Clockworkers holds their yearly meeting in the halls of a tower.
As it is tradition, after long and strenuous negotiations, the attendees take part in a curious activity. Every minute, a clockworker leaves the building and starts walking around the tower with a constant pace.
They keep on lapping turning around the tower, until everyone reaches the gate at the exact same instant.
This year's participants finish laps in 3, 5, 7 and 8 minutes, in leaving order. How many minutes will pass until they are ready to go home?
Hint
Notice that 3, 5, 7 and 8 pair-wise coprime.
Bonus
Suppose that 100 participants leave, such that the n-th clockworker walks around the tower with a period equal to the n-th prime number: the first finishes a lap every 2 minutes; the second, every 3 minutes; and so on.
How many minutes will it take in this situation?
The 100 first prime numbers
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73
79 83 89 97 101 103 107 109 113 127 131 137 139 149 151
157 163 167 173 179 181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277 281 283 293 307 311
313 317 331 337 347 349 353 359 367 373 379 383 389 397
401 409 419 421 431 433 439 443 449 457 461 463 467 479
487 491 499 503 509 521 523 541