Number of natural number 'n' for which \frac{{{n^2} + n + 12}}{n} is a natural number, is

To find the number of natural numbers \( n \) for which the expression

\[
\frac{n^2 + n + 12}{n}
\]

is a natural number, let's simplify the given expression first.

1. **Simplify the Expression:**

   The given expression is:

   \[
   \frac{n^2 + n + 12}{n}
   \]

   We can separate this into two parts:

   \[
   \frac{n^2}{n} + \frac{n}{n} + \frac{12}{n}
   \]

   Simplifying further:

   \[
   n + 1 + \frac{12}{n}
   \]

2. **Conditions for the Expression to be a Natural Number:**

   For the entire expression to be a natural number, \(\frac{12}{n}\) must also be a natural number. This implies that \( n \) should be a divisor of 12.

3. **Finding the Divisors of 12:**

   The divisors of 12 are the natural numbers that divide 12 without leaving a remainder. These divisors are:

   \[
   1, 2, 3, 4, 6, 12
   \]

4. **Checking the Values of \( n \):**

   Let's check each divisor to see if it makes the expression a natural number:

   - For \( n = 1 \):
     
     \[
     n + 1 + \frac{12}{n} = 1 + 1 + \frac{12}{1} = 1 + 1 + 12 = 14
     \]

   - For \( n = 2 \):
     
     \[
     n + 1 + \frac{12}{n} = 2 + 1 + \frac{12}{2} = 2 + 1 + 6 = 9
     \]

   - For \( n = 3 \):
     
     \[
     n + 1 + \frac{12}{n} = 3 + 1 + \frac{12}{3} = 3 + 1 + 4 = 8
     \]

   - For \( n = 4 \):
     
     \[
     n + 1 + \frac{12}{n} = 4 + 1 + \frac{12}{4} = 4 + 1 + 3 = 8
     \]

   - For \( n = 6 \):
     
     \[
     n + 1 + \frac{12}{n} = 6 + 1 + \frac{12}{6} = 6 + 1 + 2 = 9
     \]

   - For \( n = 12 \):
     
     \[
     n + 1 + \frac{12}{n} = 12 + 1 + \frac{12}{12} = 12 + 1 + 1 = 14
     \]

All the values for the divisors of 12 make the expression a natural number.

5. **Conclusion:**

   The number of natural numbers \( n \) for which the given expression is a natural number is equal to the number of divisors of 12. 

   Since 12 has 6 divisors, there are **6 natural numbers** for which the given expression is a natural number.

\[
\boxed{6}
\]

Would you like more details or have any questions?

### Related Questions

1. How do you find the divisors of a given number?
2. What happens if we replace the constant 12 with another number in the expression?
3. Can we generalize this result for any expression of the form \(\frac{n^2 + n + k}{n}\)?
4. What are the properties of divisors of a number that are useful in such problems?
5. How do we determine if an expression is a natural number in general?

### Tip

When faced with a fraction that needs to be a natural number, check if the denominator divides the numerator without leaving a remainder. This ensures the result is a whole number.

Find out how many natural numbers satisfy the condition that the expression (n^2 + n + 12)/n is a natural number. This problem involves simplifying the expression, finding divisors, and checking conditions for natural numbers. Suitable for Grades 7-9 mathematics enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation