## Prime Num3ers| (a+b+c) and (a*b*c) end with digit 7

Goal:

To find three different prime numbers   A, B, C   so that their sum   (A + B + C)   and their product   A*B*C   end with the digit 7

For example,

3  +  17  +  17  =  37                                                 3  *  17  *  17  =  867
7  +  13  +  17  =  37                                                 7  *  13  *  17  =  1547

Find other solutions.

All the prime numbers under 1000:

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, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

If the first term,   A = 3,   then   B   and   C   need to end with the digit 7:

Here are more examples:

3 + 7 + 37 = 47                        3 * 7 * 37 = 777
3 + 17 + 37 = 57                      3 * 17 * 37 = 1887
3 + 37 + 47 = 87                     3 * 37 * 47 = 5217
3 + 47 + 67 = 117                   3 * 47 * 67 = 9447
3 + 67 + 97 = 167                   3 * 67 * 97 = 19497
3 + 97 + 107 = 207               3 * 97 * 107 = 31137
3 + 107 + 127 = 237             3 * 107 * 127 = 40767
3 + 127 + 137 = 267             3 * 127 * 137 = 52197
3 + 137 + 157 = 297             3 * 137 * 157 = 64527
3 + 157 + 167 = 327             3 * 157 * 167 = 78657
3 + 167 + 197 = 367             3 * 167 * 197 = 98697
3 + 197 + 227 = 427            3 * 197 * 227 = 134157
3 + 227 + 257 = 487            3 * 227 * 257 = 175017
3 + 257 + 277 = 537            3 * 257 * 277 = 213567
3 + 277 + 307 = 587            3 * 277 * 307 = 255117

Can   A   =   B   =   3 ?
What if   A ≠ 3 ? 1. themusegarden says:
–moreover, (this probably doesn’t cover all the cases), but a combination of primes ending in 3, 7, 7 (as in your example) will provide the ending digit 7 for both addition and multiplication. $3+7+7=17$ and $3*7*7=147$, both of which end in 7.
–for addition, any combination of a pair whose ones add to ten and a 7 would do the trick (e.g., $1+9+7=17$) however, $1*9*7=63$ which does not end in 7.