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 equation whose roots are p and q, is

Options

A. `x^2-3x+2=0`
B. `x^2-5x+6=0`
C. `x^2-7x+12=0`
D. `x^2-9x+20=0`

Answer

To solve the problem, we need to find the values of \( U \) and \( M \) and then compute the fraction \( \frac{U}{M} \). Finally, we will express this fraction in its simplest form and find the equation whose roots are \( p \) and \( q \).

### Step 1: Calculate \( U \)
The number of digits \( U \) in a number \( n \) can be found using the formula:
\[ U = \lfloor \log_{10} n \rfloor + 1 \]
For \( n = 60^{100} \):
\[ U = \lfloor \log_{10} (60^{100}) \rfloor + 1 = \lfloor 100 \log_{10} 60 \rfloor + 1 \]
Using the property of logarithms:
\[ \log_{10} 60 = \log_{10} (6 \times 10) = \log_{10} 6 + 1 \]
Next, we need to calculate \( \log_{10} 6 \):
\[ \log_{10} 6 = \log_{10} (2 \times 3) = \log_{10} 2 + \log_{10} 3 \]
Given \( \log_{10} 2 = 0.301 \) and \( \log_{10} 3 = 0.477 \):
\[ \log_{10} 6 = 0.301 + 0.477 = 0.778 \]
Now substituting back:
\[ \log_{10} 60 = 0.778 + 1 = 1.778 \]
Now calculate \( U \):
\[ U = \lfloor 100 \times 1.778 \rfloor + 1 = \lfloor 177.8 \rfloor + 1 = 177 + 1 = 178 \]

### Step 2: Calculate \( M \)
The number of digits after the decimal point before a significant figure in \( 8^{-296} \) can be calculated similarly:
\[ M = \lfloor -\log_{10} (8^{-296}) \rfloor \]
This simplifies to:
\[ M = \lfloor 296 \log_{10} 8 \rfloor \]
Using the property of logarithms:
\[ \log_{10} 8 = \log_{10} (2^3) = 3 \log_{10} 2 = 3 \times 0.301 = 0.903 \]
Now substituting back:
\[ M = \lfloor 296 \times 0.903 \rfloor = \lfloor 267.288 \rfloor = 267 \]

### Step 3: Calculate \( \frac{U}{M} \)
Now we have \( U = 178 \) and \( M = 267 \):
\[ \frac{U}{M} = \frac{178}{267} \]
To simplify \( \frac{178}{267} \), we find the greatest common divisor (GCD) of 178 and 267, which is 89:
\[ \frac{178 \div 89}{267 \div 89} = \frac{2}{3} \]
Thus, \( p = 2 \) and \( q = 3 \).

### Step 4: Find the equation with roots \( p \) and \( q \)
The equation whose roots are \( p \) and \( q \) can be written as:
\[ (x - p)(x - q) = 0 \]
Substituting \( p \) and \( q \):
\[ (x - 2)(x - 3) = 0 \]
Expanding this gives:
\[ x^2 - 5x + 6 = 0 \]

### Final Answer
The equation whose roots are \( p \) and \( q \) is:
\[ \boxed{x^2 - 5x + 6 = 0} \]