Squares in different bases

Question

In base 5, 121 is 25+10+1=36=6*6

In base 6 121 is 36+12+1=49=7*7

In base 7 121 is 49+14+1=64=8*8

In base 5 144 is 25+20+4=49=7*7

In base 6 144 is 36+24+4=64=8*8

Can someone please explain why there seems to be this pattern? Thanks

Rik

Answer 1

Working in any base $b$ (assuming $b>2$), we have $$ 11^2=(10+1)^2=100+20+1 $$ And, assuming $b>4$, we have $$ 12^2=(10+2)^2=100+40+4 $$ Similarly, whenever $b>9$, you are going to find $$ 13^2=169 $$