Suppose `U` denotes the number of digits in the number` (60)^(100)` and`M` denotes the number of cyphers after decimal, before a significant figure comes in `(8)^(-296)`. If the fraction U/M is expressed as rational number in the lowest term as `p//q` (given `log_(10)2=0.301` and `log_(10)3=0.477`) .
The value of p is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will first find the values of \( U \) and \( M \), and then calculate the fraction \( \frac{U}{M} \) in its lowest terms.

### Step 1: Calculate \( U \)
\( U \) denotes the number of digits in the number \( (60)^{100} \).

The number of digits \( d \) in a number \( n \) can be found using the formula:
\[ d = \lfloor \log_{10} n \rfloor + 1 \]
Thus, we have:
\[ U = \lfloor \log_{10} (60^{100}) \rfloor + 1 \]
Using the logarithmic property \( \log_{10} (a^b) = b \cdot \log_{10} a \), we can write:
\[ U = \lfloor 100 \cdot \log_{10} 60 \rfloor + 1 \]

Next, we need to find \( \log_{10} 60 \):
\[ \log_{10} 60 = \log_{10} (6 \times 10) = \log_{10} 6 + \log_{10} 10 \]
Since \( \log_{10} 10 = 1 \), we can express \( \log_{10} 6 \) as:
\[ \log_{10} 6 = \log_{10} (2 \times 3) = \log_{10} 2 + \log_{10} 3 \]
Substituting the values:
\[ \log_{10} 2 = 0.301 \quad \text{and} \quad \log_{10} 3 = 0.477 \]
Thus:
\[ \log_{10} 6 = 0.301 + 0.477 = 0.778 \]
Now, substituting back:
\[ \log_{10} 60 = 0.778 + 1 = 1.778 \]
Now we can find \( U \):
\[ U = \lfloor 100 \cdot 1.778 \rfloor + 1 = \lfloor 177.8 \rfloor + 1 = 177 + 1 = 178 \]

### Step 2: Calculate \( M \)
\( M \) denotes the number of ciphers after the decimal in \( (8)^{-296} \).

We can express this as:
\[ M = \text{number of ciphers after decimal in } (8)^{-296} = \text{number of digits in } (8)^{296} \]
This can be calculated similarly:
\[ M = \lfloor \log_{10} (8^{-296}) \rfloor + 1 \]
Using properties of logarithms:
\[ M = \lfloor -296 \cdot \log_{10} 8 \rfloor + 1 \]
Now, we need to find \( \log_{10} 8 \):
\[ \log_{10} 8 = \log_{10} (2^3) = 3 \cdot \log_{10} 2 = 3 \cdot 0.301 = 0.903 \]
Substituting back:
\[ M = \lfloor -296 \cdot 0.903 \rfloor + 1 = \lfloor -267.288 \rfloor + 1 = -268 + 1 = 267 \]

### Step 3: Calculate \( \frac{U}{M} \)
Now we can find the fraction:
\[ \frac{U}{M} = \frac{178}{267} \]

### Step 4: Simplify \( \frac{U}{M} \)
To simplify \( \frac{178}{267} \), we find the GCD of 178 and 267. The GCD is 89:
\[ \frac{178 \div 89}{267 \div 89} = \frac{2}{3} \]

### Final Result
Thus, \( \frac{U}{M} = \frac{2}{3} \), and the value of \( p \) is:
\[ \boxed{2} \]