MATH with my KIDS

Synchronicity

I first heard the word synchronicity when I was sitting in an airport, waiting for a plane, working on this math problem (for fun) and over-hearing a man’s cell phone conversation. I thought that it was a good word for this phenomenon. Here it is:

Have you noticed that 3 times 7 is 21 and if you expand 7 base 3 it is 21. That is:

$21=3\times7$

$21_3=7$

This seems like an interesting coincidence. Could it be that this happens for any other numbers? Maybe it happens infinitely often, maybe never. Asking such questions is where the adventure begins. Of course we are implicitly working base 10 here. That is, in the first line the string of numerals “21″ is interpreted as a base 10 number. In the second one it is (explicitly) interpreted base 3. You could allow 10 to be replace by other bases also.

If we restrict our attention to base 10 (in the first line), I claim that, by exhaustive computer search, there are only five other instances of this phenomenon. They are:

$1,\!066,\!338,\!805,\!156,\!287,\!287,\!067$ $=9\times118,\!482,\!089,\!461,\!809,\!698,\!563$ and $1,\!066,\!338,\!805,\!156,\!287,\!287,\!067_9$ $=118,\!482,\!089,\!461,\!809,\!698,\!563$

$1,\!124,\!161,\!329,\!714,\!632,\!881,\!704$ $=9\times124,\!906,\!814,\!412,\!736,\!986,\!856$ and $1,\!124,\!161,\!329,\!714,\!632,\!881,\!704_9$ $=124,\!906,\!814,\!412,\!736,\!986,\!856$

$2,\!305,\!867,\!155,\!177,\!711,\!644,\!802$ $=9\times256,\!207,\!461,\!686,\!412,\!404,\!978$ and $2,\!305,\!867,\!155,\!177,\!711,\!644,\!802_9$ $=256,\!207,\!461,\!686,\!412,\!404,\!978$

$2,\!306,\!166,\!776,\!784,\!312,\!535,\!170$ $=9\times256,\!240,\!752,\!976,\!034,\!726,\!130$ and $2,\!306,\!166,\!776,\!784,\!312,\!535,\!170_9$ $=256,\!240,\!752,\!976,\!034,\!726,\!130$

$5,\!744,\!341,\!611,\!556,\!736,\!174,\!883$ $=9\times638,\!260,\!179,\!061,\!859,\!574,\!987$ and $5,\!744,\!341,\!611,\!556,\!736,\!174,\!883_9$ $=638,\!260,\!179,\!061,\!859,\!574,\!987$