2-digit numbers:
Solutions: a = 0, b = 0 & a = 2, b = 2
22 |—–> 2 + 2 = 4 = 2 * 2
3-digit numbers:
123 |—–> 1 + 2 + 3 = 6 = 1 * 2 * 3
Also true for 132, 213, 231, 312, 321
4-digit numbers:
1124 |—–> 1 + 1 + 2 + 4 = 8 = 1 * 1 * 2 * 4
1142 |—–> 1 + 1 + 4 + 2 = 8 = 1 * 1 * 4 * 2
And other combinations of 1, 1, 2 and 4
5-digit numbers:
11125 |—–> 1 * 1 * 1 * 2 * 5 = 10 = 1 + 1 + 1 + 2 + 5
11152 |—–> 1 * 1 * 1 * 5 * 2 = 10 = 1 + 1 + 1 + 5 + 2
And other combinations of 1, 1, 1, 2 and 5
11133 |—–> 1 * 1 * 1 * 3 * 3 = 9 = 1 + 1 + 1 + 3 + 3
And other combinations of 1, 1, 1, 3 and 3
11222 |—–> 1 * 1 * 2 * 2 * 2 = 8 = 1 + 1 + 2 + 2 + 2
And other combinations of 1, 1, 2, 2 and 2
6-digit numbers:
111126 |—–> 1 * 1 * 1 * 1 * 2 * 6 = 12 = 1 + 1 + 1 + 1 + 2 + 6
111162 |—–> 1 * 1 * 1 * 1 * 6 * 2 = 12 = 1 + 1 + 1 + 1 + 6 + 2
And other combinations of 1, 1, 1, 1, 6 and 2
112411 |—–> 1 * 1 * 2 * 4 * 1 * 1 = 8 = 1 + 1 + 2 + 2 + 1 + 1
And other combinations of 1, 1, 2, 4, 1 and 1
7-digit numbers:
1111127 |—–> 1 * 1 * 1 * 1 * 1 * 2 * 7 = 14 = 1 + 1 + 1 + 1 + 1 + 2 + 7
1111172 |—–> 1 * 1 * 1 * 1 * 1 * 7 * 2 = 14 = 1 + 1 + 1 + 1 + 1 + 7 + 2
And other combinations of 1, 1, 1, 1, 1, 2 and 7
1111134 |—–> 1 * 1 * 1 * 1 * 1 * 3 * 4 = 12 = 1 + 1 + 1 + 1 + 1 + 3 + 4
1111143 |—–> 1 * 1 * 1 * 1 * 1 * 4 * 3 = 12 = 1 + 1 + 1 + 1 + 1 + 4 + 3
And other combinations of 1, 1, 1, 1, 1, 3 and 4
8-digit numbers:
11111128 |—–> 1 * 1 * 1 * 1 * 1 * 1 * 2 * 8 = 16 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 8
11111182 |—–> 1 * 1 * 1 * 1 * 1 * 1 * 2 * 8 = 16 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 8
And other combinations of 1, 1, 1, 1, 1, 1, 2 and 8
9-digit numbers:
111111129 |—–> 1 * 1 * 1 * 1 * 1 * 1 * 1 * 2 * 9 = 18 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 + 9
111111192 |—–> 1 * 1 * 1 * 1 * 1 * 1 * 1 * 9 * 2 = 18 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 9 + 2
And other combinations of 1, 1, 1, 1, 1, 1, 1, 2 and 9
111111135 |—–> 1 * 1 * 1 * 1 * 1 * 1 * 1 * 3 * 5 = 15 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 3 + 5
111111153 |—–> 1 * 1 * 1 * 1 * 1 * 1 * 1 * 5 * 3 = 15 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 5 + 3
And other combinations of 1, 1, 1, 1, 1, 1, 1, 3 and 5
10-digit numbers:
1*1*1*1*1*1*1*1*x*y = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + x + y
x * y = 8 + x + y, with 0 ≤ x, y ≤ 9
so I get x = y = 4
1111111144 |—–> 1*1*1*1*1*1*1*1*4*4 = 16 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 4 + 4
And other combinations of 1, 1, 1, 1, 1, 1, 1, 1, 4 and 4
Question: Generalize this.
