Logarithms

Since $0 \le \beta_1, \beta_2 < 1,$ we can rewrite the given system of equations as $\begin{aligned} 4 &\le \log_3 N < 5 \\ 2 &\le \log_5 N < 3. \end{aligned}$ Taking note that logarithm functions are increasing, we can convert these as follows: $\begin{aligned} 3^4 &\le N < 3^5 \\ 5^2 &\le N < 5^3, \end{aligned}$ or $\begin{aligned} 81 &\le N < 243 \\ 25 &\le N < 125.\end{aligned}$ $N$ must satisfy both of these inequality chains, so we get $81 \le N < 125,$ or $81 \le N \le 124.$ Thus, the possible values of $N$ are just the integers between $81$ and $124$ inclusive, of which there are $124-81+1 = \boxed{44}.$

By law of logarithms and exponents:

the first equation is equivalent to

$\large 3^{4 + \beta_1} = N$

at the same time, the second equation is also equivalent to

$\large 5^{2 + \beta_2} = N$

Since,

• $\beta_1$ and $\beta_2$ are fractional parts of a number and they are between 0 and 1

• $3^4 = 81$ and $3^5 = 243$

• $5^2 = 25$ and $5^3 = 125$

the number of "cool numbers" is the intersection of the numbers in my last two conditions. The intersection will be the numbers between 81 and 125. Therefore, the number of "cool numbers" is $125 - 81$ or 44

converting log into exponential form N=3^{4+B1}=>is in between [81,243) N=5^{2+B2}=>is in b/n [25,125) taking intersection answer is \boxed{44}

\­[ N=3^{4} 3^{\beta {1}}\­] and [N=5^{2} 5^{\beta {2}} The range of N are N=[81 243) and N=[25 125) form both the equation hence the commen area is [81 124] hence 44 integer

From the equations, we know $\lfloor\log_3 N\rfloor=4$ and $\lfloor\log_5 N\rfloor=2$ From the first equation, $N\ge 3^4=81$ and $N<3^5=243$ . From the second, $N\ge 5^2=25$ and $N<5^3=125$ . Therefore, the answer is the number of integers from 81 to 124, inclusive, which is $124-80=\boxed{44}$ .

$\log_3 N = 4 + \beta_1$ is another way to say that $\log_3 N$ lies in the interval $[4,5)$ meaning that $N \in [81,243)$ . Also $\log_5 N = 2 + \beta_2$ means that $\log_5 N \in [2,3)$ and so $N \in [25,125)$ .

We are left with $N \in [81,125)$ . This interval has $44$ integers.