Word problem question

Question

A three-digit number is the product of four prime numbers. Given that the three digits of the number are all prime and different and that the sum of its prime factors is 22, find the number.

Answer 1

2,3,5,7 are single digit primes

235 ---> 235 = 5*47 NO

237 --> 237 = 3 * 79 NO

257 is prime

357 ---> 3 * 119 = 3 * 7 * 17

These are the 4 ways to choose 3 out of 4.

But the digits in each of these 3 digit numbers can be rearranged.

First, 235:

253 ---> 253 = 11 * 23 NO

325 ---> 325 = 25 * 13 = 5 * 5 * 13 NO

352 ---> 352 = 2 * 176 = 2 * 2 * 88 = 2*2*2*2*2*11 NO

532 --> 532 = 2 * 266 = 2 * 2 * 133 = 2 * 2 * 7 * 19 NO

523 is prime

Next, 237

273 ---> 273 = 3 * 91 = 3 * 7 * 13 NO

723 ---> 723 = 3 * 241 NO

732 ---> 732 = 2 * 366 = 2 * 2 * 183 = 2 * 2 * 3 * 61 NO

327 ---> 327 = 3 * 109 NO

372 ---> 372 = 3 * 124 = 3 * 2 * 62 = 3 * 2 * 2 * 31 NO

Next , 257

275 --> 275 = 25 * 11 = 5 * 5 * 11 NO

572 --> 572 = 2 * 286 = 2 * 2 * 143 = 2 * 2 * 11 * 13 NO

527 ---> 527 = 17 * 31 NO

752 ---> 752 = 2 * 376 = 2 * 2 * 188 = 2*2*2*94 = 2*2*2*2*47 NO

725 ---> 725 = 25 * 29 = 5*5*29 NO

Last 357

375 ---> 375 = 25 * 15 = 5*5*3*5 NO

735 --> 735=5 * 147 = 5 * 3 * 49 = 5 * 3 * 7 * 7 YES!!! the total is 22

735 = 3 * 5 * 7 * 7