Factor 1

Lets start with the number 145,277

In binary this would be written 100011011101111101

reading from the end, the last digit represents the number 1

there is no 2

the next digit represents 4

followed by an 8

followed by a 16

notice how the number keeps doubling

the number 145277 therefore = 131072 + 0 + 32768 + 16384 + 8192 + 4096 + 2048 + 0 + 512 + 256 + 128 + 0 + 32 + 16 + 0 + 0 + 0 + 1
                                                     1         0         1            1           1           1          1       0       1         1       1      0      1       1    0    0     0    1

The number is placed on the board with the smallest number at the top and the largest at the bottom

The light coloured blocks can only be moved horizontally across the board. The aim is to create the same arrangement of blocks in all columns.

We now have created 4 columns with 3 blocks in each.

Counting down the first column we have 1 + 4 + 512 = 517

Counting down the outer edge diagonal we have 1 + 8 + 16 + 256 = 281

517 and 281 are both factors of the number 145277

So by finding the pattern we have been able to find the factors of the number.

Lets try a really big number : 5357543035931336604742125245300009052807024058527668037218751941851755255624680612465991894078479290
                                            6379733645877657341259357264287040473037870728192308072837957801638409157897297547515241043699508846
                                            12523266964540523800142685651570380681535500510599629488563458108256606374668096361912712465966178593
This number is 301 digits long and = (2^999)+(2^566)+(2^558)+(2^446)+(2^441)+(2^13)+(2^8)+(2^5)+(2^0)

Placed into the grid it would look like the diagram below

Of course the grid is not to scale as it would have to be 1001 rows high.

The divisors of the 301 digit number are (2^558)+(2^5)+(2^0) = 94349060620538533806038864524706722272923030510411010709405157
                                                                                                   50614060405980372130215316812944146918853670937576909612249426
                                                                                                   46157481198158140358562858174010912348831777

and (2^441)+(2^8)+(2^0) = 56784275335594288324165922491250354246378231303696723459491421 81098
                                             744438385921275985867583701277855943457200048954515105739075223809