79
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We start with the left end of the number 553. Since 7 is larger than 5, we cannot get any 7's from the hundreds column, so we move to the next column. I know that 7 x 7 = 49, therefore we can get seven sevens out of 55. And since it is taken from the tens column, it's actually 70 sevens. We can't get eight sevens because 8 x 7 = 56 and 56 is larger than 55. So, if we subtract the 49 from the 55 we have 6 left over from the hundreds and tens columns combined (which is actually 60 left over, since it is in the tens column). Now we have one more column which contains the number 3, and 60 + 3 = 63, so we have 63 left over, in total. We can get nine sevens out of 63 because 9 x 7 = 63. So, therefore the complete answer is, 553/7 = 79.
553
9 times, with 13 left over
The rule being used here is probably a recursive one: xn+2 = xn + xn+1 + 2 For instance, 48 = 17 + 29 + 2. Thus the nextnumber is: 29 + 48 + 2 = 79 And the series can be extended indefinitely in both directions, if we like: ...-39, 21, -16, 7, -7, 2, -3, 1, 0, 3, 5, 10, 17, 29, 48, 79, 129, 210, 341, 553...
There are infinitely many multiples of 7, they start: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343, 350, 357, 364, 371, 378, 385, 392, 399, 406, 413, 420, 427, 434, 441, 448, 455, 462, 469, 476, 483, 490, 497, 504, 511, 518, 525, 532, 539, 546, 553, 560, 567, 574, 581, 588, 595, 602, 609, 616, 623, 630, 637, 644, 651, 658, 665, 672, 679, 686, 693, 700, ...
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343, 350, 357, 364, 371, 378, 385, 392, 399, 406, 413, 420, 427, 434, 441, 448, 455, 462, 469, 476, 483, 490, 497, 504, 511, 518, 525, 532, 539, 546, 553, 560, 567, 574, 581, 588, 595, 602, 609, 616, 623, 630, 637, 644, 651, 658, 665, 672, 679, 686, 693, 700, 707, 714, 721, 728, 735, 742, 749, 756, 763, 770, 777, 784, 791, 798, 805, 812, 819, 826, 833, 840, 847, 854, 861, 868, 875, 882, 889, 896, 903, 910, 917, 924, 931, 938, 945, 952, 959, 966, 973, 980, 987, 994.