Riemann's green is a green pigment ZnO.CoO, which is obtained by heating a mixture of zinc oxide and cobalt nitrate or by the reaction of cobalt nitrate with either 3Zn(OH)2ZnCO3 or zinc sulphate.

Log_2aa=x    then, a=(2a)^x  ......(1)

Log_3a2a=y    then,2a=(3a)^y ......(2)

Log_4a 3a=z  then, 3a=(4a)^z ......(3)

So, 

a=(2a)^x  [from (1)]

Or, a=(3a)^xy    [from(2)]

Or, a=(4a)^xyz     [from(3)]

Multiplying both sides by 4a,

4a.a=4a.(4a)^xyz  

Or,(2a)² =(4a)^{xyz + 1} 

Or,(3a)^2y =(4a)^xyz+1 

Or,(4a)^2yz =(4a)^xyz+1 

Or, 2yz = xyz+1 .proved.

Yes, a physical quantity can have magnitude and direction but still be a scalar if it doesn't obey the vector addition. An example is Electric Current which has magnitude and a fixed direction, but it does not follow vector laws of addition.

The stage of cell division in which paired homologous chromosomes get shortened and thickened is prophase I of meiosis. During this stage, the chromosomes condense and become visible as distinct structures, allowing for the homologous chromosomes to pair up and exchange genetic material through a process called crossing over. The shortening and thickening of the chromosomes during prophase I is important for proper alignment and separation of the homologous chromosomes during subsequent...