How many $4$ Digit numbers are there in the decimal system, which have exactly $4$ Digits when expressed in Base $6$, Base $7$, Base $8$?

Find the number of solutions of the equation $\left(x^2-y^2\right)=777314$.

There was a table in a room and there were $100$ coins ( coin $1$ to coin $100$) in a row on the table. All the coins were heads up initially. You entered the room and turned all the coins, the second time you entered the room and turned every $2$nd coin (coin $2$, coin $4$, .....), the $3$rd time you entered the room and turned every $3$rd coin ( coin $3$, coin $6$, coin $9$, .....) and so on, if you visited the room $100$ times and continued this sequence every time. Then answer the following question

What were the states of $54$th and $91$th coin after the $100$th visit? Type $1$ if both are heads, Type $2$ if the first one is head and the other is tail, Type $3$ if the first one is tail and the second is tail, Type $4$ if both are tails.