If characteristic of three numbers a, b and c and 5, -3 and 2, respectively, then find the maximum number of digits in N = abc.

Explanation

To find the maximum number of digits in \( N = abc \) given the characteristics of the numbers \( a, b, \) and \( c \) are 5, -3, and 2 respectively, we can follow these steps:

### Step 1: Understand the Characteristics
The characteristic of a number in logarithmic terms indicates how many digits are to the left of the decimal point. For example, a characteristic of 5 means the number is between \( 10^5 \) and \( 10^6 \).

### Step 2: Express the Logarithms
We can express the logarithms of the numbers as follows:
- \( \log a = 5 + p \) (where \( 0 \leq p < 1 \))
- \( \log b = -3 + q \) (where \( 0 \leq q < 1 \))
- \( \log c = 2 + r \) (where \( 0 \leq r < 1 \))

### Step 3: Calculate \( \log N \)
Using the property of logarithms that states \( \log(mn) = \log m + \log n \), we can find \( \log N \):
\[ \log N = \log(abc) = \log a + \log b + \log c \]
Substituting the expressions from Step 2:
\[ \log N = (5 + p) + (-3 + q) + (2 + r) \]
Simplifying this:
\[ \log N = 5 - 3 + 2 + p + q + r = 4 + p + q + r \]

### Step 4: Determine the Range of \( p + q + r \)
Since \( p, q, r \) are all in the range [0, 1), the maximum value of \( p + q + r \) can be:
\[ p + q + r < 3 \]
Thus, the maximum possible value of \( p + q + r \) is just below 3.

### Step 5: Find the Maximum Value of \( \log N \)
The maximum value of \( \log N \) can be approximated as:
\[ \log N < 4 + 3 = 7 \]

### Step 6: Calculate the Number of Digits
The number of digits \( d \) in a number \( N \) can be found using the formula:
\[ d = \lfloor \log N \rfloor + 1 \]
Since \( \log N < 7 \), we have:
\[ \lfloor \log N \rfloor \leq 6 \]
Thus, the maximum number of digits in \( N \) is:
\[ d = 6 + 1 = 7 \]

### Conclusion
The maximum number of digits in \( N = abc \) is **7**.