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 q is

Options

A. 5
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the values of \( U \) and \( M \) as defined in the question, and then compute the fraction \( \frac{U}{M} \).

### Step 1: Calculate \( U \)

We start by calculating \( U \), which is the number of digits in \( (60)^{100} \).

The formula to find the number of digits \( d \) in a number \( n \) is given by:
\[ d = \lfloor \log_{10} n \rfloor + 1 \]

Thus, for \( (60)^{100} \):
\[ U = \lfloor \log_{10} (60^{100}) \rfloor + 1 \]
\[ = \lfloor 100 \log_{10} 60 \rfloor + 1 \]

Next, we need to calculate \( \log_{10} 60 \):
\[ \log_{10} 60 = \log_{10} (2 \times 3 \times 10) = \log_{10} 2 + \log_{10} 3 + \log_{10} 10 \]
\[ = \log_{10} 2 + \log_{10} 3 + 1 \]

Using the given values \( \log_{10} 2 = 0.301 \) and \( \log_{10} 3 = 0.477 \):
\[ \log_{10} 60 = 0.301 + 0.477 + 1 = 1.778 \]

Now substituting back into the equation for \( U \):
\[ U = \lfloor 100 \times 1.778 \rfloor + 1 = \lfloor 177.8 \rfloor + 1 = 177 + 1 = 178 \]

### Step 2: Calculate \( M \)

Now we calculate \( M \), which is the number of ciphers after the decimal before a significant figure in \( (8)^{-296} \).

First, rewrite \( (8)^{-296} \):
\[ (8)^{-296} = \frac{1}{(8)^{296}} = 10^{-y} \]
where \( y = \log_{10} (8^{296}) \).

Calculating \( y \):
\[ y = 296 \log_{10} 8 \]
We can express \( \log_{10} 8 \) as:
\[ \log_{10} 8 = \log_{10} (2^3) = 3 \log_{10} 2 \]
Thus,
\[ y = 296 \times 3 \log_{10} 2 = 888 \log_{10} 2 \]

Substituting the value of \( \log_{10} 2 \):
\[ y = 888 \times 0.301 = 267.288 \]

Now, since \( M \) is the number of ciphers after the decimal before a significant figure:
\[ M = \lfloor y \rfloor = \lfloor 267.288 \rfloor = 267 \]

### Step 3: Calculate \( \frac{U}{M} \)

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

### Step 4: Express \( \frac{U}{M} \) in Lowest Terms

To express \( \frac{178}{267} \) in lowest terms, we check the greatest common divisor (GCD) of 178 and 267. Since 178 and 267 have no common factors other than 1, the fraction is already in its simplest form.

### Final Result

Thus, in the form \( \frac{p}{q} \), we have \( p = 178 \) and \( q = 267 \). The value of \( q \) is:

\[ \boxed{267} \]