Understanding the Problem

We need to find the number of factors of the number $N = 2^3 \times 3^3 \times 5^4 \times 7^2$ that satisfy two specific conditions: they must be divisible by 50, and they must not be divisible by 100.

Prime Factorization

The number is given as $N = 2^3 \times 3^3 \times 5^4 \times 7^2$.

Any factor of N has the general form $F = 2^a \times 3^b \times 5^c \times 7^d$, where the exponents are within the following ranges:

• $0 \le a \le 3$
• $0 \le b \le 3$
• $0 \le c \le 4$
• $0 \le d \le 2$

Applying Divisibility Conditions

Condition 1: Divisible by 50

The prime factorization of 50 is $2^1 \times 5^2$. For a factor F to be divisible by 50, its exponents must satisfy $a \ge 1$ and $c \ge 2$.

Condition 2: Not divisible by 100

The prime factorization of 100 is $2^2 \times 5^2$. For a factor F to be divisible by 100, its exponents must satisfy $a \ge 2$ and $c \ge 2$. Therefore, for F *not* to be divisible by 100, we must have $a < 2$ or $c < 2$.

Combining Conditions

We need factors that satisfy both conditions:

• Divisible by 50: $a \ge 1$ and $c \ge 2$.
• Not divisible by 100: $a < 2$ or $c < 2$.

Combining these, we need:

• ($a \ge 1$ and $c \ge 2$) AND ($a < 2$ or $c < 2$).

Since $c \ge 2$ is required by the first condition, the second condition simplifies. The combined condition becomes ($a \ge 1$ and $c \ge 2$) AND ($a < 2$).

This simplifies further to $a = 1$ and $c \ge 2$.

Determining Exponent Ranges

Now, let's find the possible ranges for the exponents $a, b, c, d$ that meet all requirements:

• For $a$: Must be exactly 1. Range is $a = 1$. Number of possibilities = 1.
• For $b$: Must be $0 \le b \le 3$. Number of possibilities = $3 - 0 + 1 = 4$.
• For $c$: Must satisfy $c \ge 2$ (from divisibility by 50) and the original range $0 \le c \le 4$. So, the range is $2 \le c \le 4$. Number of possibilities = $4 - 2 + 1 = 3$.
• For $d$: Must be $0 \le d \le 2$. Number of possibilities = $2 - 0 + 1 = 3$.

Calculating the Number of Factors

The total number of factors satisfying the conditions is the product of the number of possibilities for each exponent:

Number of factors = (Possibilities for $a$) $\times$ (Possibilities for $b$) $\times$ (Possibilities for $c$) $\times$ (Possibilities for $d$)

Number of factors = $1 \times 4 \times 3 \times 3 = 36$.

Conclusion

There are 36 factors of $2^3 \times 3^3 \times 5^4 \times 7^2$ that are divisible by 50 but not by 100.