Convert them into an algebraic expression first!

$\underbrace{16 \div 16 \div 16 \div \cdots \div 16 }_{\text{number of 16's }=\, N} = 16 \div \left( \underbrace{ 16 \times 16\times \cdots \times 16 }_{\text{number of 16's }=\, N-1}\right) = 16 \div 16^{N-1} = 16^{1- (N-1)} = 16^{2-N}.$

Similarly,

$\underbrace{64 \div 64 \div 64 \div \cdots \div 64 }_{\text{number of 64's }=\, 8} =64\div \left( \underbrace{ 64\times 64 \times \cdots \times 64 }_{\text{number of 64's }=\, 7} \right) = 64 \div 64^7 = 64^{1-7} = 64^{-6}.$

Equating these two gives

$\begin{aligned} 16^{2-N} &= &64^{-6} \\ (4^2)^{2-N} &= &(4^3)^{-6} \\ 4^{2(2-N)} &= &4^{3(-6)} \\ 2(2-N) &= & 3(-6) \\ 4 - 2N &= & - 18 \\ -2N &= & -22 \\ N&= & \boxed{11} . \end{aligned}$

$(16÷16)÷16...÷16÷16 = (64 ÷64)÷64÷64÷64÷64÷64÷64$

$1 ÷ 16^{N - 2} = 1 ÷ 64^6$

$16^{2 - N} = 64^ {-6}$

$\text {Since } 16 = 4^2 \text { and } 64 = 4^3 \text { , therefore: }$

$4^{2(2 - N)} = 4^{3 × (-6)}$

$4 - 2N = -18$

$2N = 22$

$N = \boxed {11}$

If there are 8 64s then that means that:

64÷64÷64÷....÷64= 64/(64^(7)) = 64^(-6)

In the same sense, if there are N number of 16s then that means that:

16÷16÷16÷....÷16= 16/[16^(N-1)]

= 16^(2-N)

so if 64^(-6) = 16^(2-N);

then [64^(-6)]^(-1)=[16^(2-N)]^(-1)

64^(6)=16^(N-2)

(4^(3))^(6)= (4^2)^(N-2)

4^(18) = 4^(2N-4)

ln[4^(18)] = ln[4^(2N-4)]

(18) ln [4] = (2N-4)ln[4]

(18) ln [4]/ln[4] = (2N-4)ln[4]/ln[4]

18 = 2N-4

18+4 = 2N

22=2N

N = 11