The number N is a multiple of 7. The base 2 representation of N is 10110101010101ABC110. Compute the ordered triple of digits (A,B,C).

Question

anonymous · Answer

this can't be that bad
if they are all ones it is $742726$

anonymous · Answer

$742782$ my mistake the first one is all zeros

anonymous · Answer

thanks!

anonymous · Answer

well consider $2^3=8\equiv1\pmod 7$ hence we can split into three-bit chunks and sum those:$$010110101010101ABC110_2\\ \ \ \equiv010_2+011_2+110_2+101_2+010_2+101_2+ABC_2+110_2\pmod 7$$

anonymous · Answer

oops the second chunk i.e. $011_2$ shouldn't be there. anywaysnow just consider:$$010_2=2\110_2=6\101_2=5$$hence we have:$$010_2+110_2+101_2+010_2+101_2+ABC_2+110_2\\ \ \ \equiv2+6+5+2+5+ABC_2+6\\ \ \ \equiv5+ABC_2\pmod 7$$considering we want the whole thing $\equiv 0\pmod 7$ observe:$$5+ABC_2\equiv0\pmod7\ABC_2\equiv-5\equiv2\pmod 7$$i.e. $ABC_2=010_2$