Ian draws triangles whose two side lengths are 33 and 55. The length of the third side
is an integer with exactly 3 factors. How many different triangles can Ian draw?

To determine how many different triangles Ian can draw, we need to consider the constraints of the triangle inequality and the condition that the third side \( c \) is an integer with exactly 3 factors.

### Step 1: Triangle Inequality Conditions

The triangle inequality states that for any triangle with sides \( a \), \( b \), and \( c \):
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In this case, let \( a = 33 \) and \( b = 55 \). The inequalities become:

1. \( 33 + 55 > c \) \\
   \( 88 > c \) \\
   Therefore, \( c < 88 \)

2. \( 33 + c > 55 \) \\
   \( c > 22 \)

3. \( 55 + c > 33 \) \\
   This is always true since \( c \) will be positive.

From these inequalities, we can combine the results:
\[ 22 < c < 88 \]

### Step 2: Finding Integer Values of \( c \) with Exactly 3 Factors

An integer has exactly 3 factors if it is the square of a prime number. This is because the number of factors \( d(n) \) of a number \( n = p^k \) (where \( p \) is prime and \( k \) is a positive integer) is given by \( d(n) = k + 1 \). Thus, for \( d(n) = 3 \), we have \( k = 2 \), which means \( n = p^2 \).

Next, we need to find the prime numbers whose squares lie within the bounds \( 22 < c < 88 \).

### Step 3: Identify Primes and Their Squares

Let's consider the primes up to \( \sqrt{88} \approx 9.38 \):
- The primes are 2, 3, 5, and 7.

Now, we compute their squares:
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 5^2 = 25 \)
- \( 7^2 = 49 \)

Now we check which of these squares fall between 22 and 88:
- \( 25 \) (valid)
- \( 49 \) (valid)

### Step 4: Valid Values for \( c \)

The valid integers for \( c \) that are squares of primes and satisfy the triangle inequalities are:
- \( 25 \)
- \( 49 \)

### Conclusion

Thus, there are **2 different triangles** that Ian can draw with sides \( 33 \), \( 55 \), and a third side \( c \) that is either \( 25 \) or \( 49 \).

Would you like further details or have any questions about the solution? Here are some related questions you might consider:

1. What are the properties of triangles that satisfy specific inequalities?
2. How can we determine the number of factors of a given integer?
3. What is the significance of prime numbers in number theory?
4. Can you provide examples of other integers that have exactly three factors?
5. How does the triangle inequality apply to other shapes, like quadrilaterals? 

**Tip:** Always verify that your integer solutions meet any defined constraints, like the triangle inequalities, before concluding.

Ian draws triangles whose two side lengths are 33 and 55. The length of the third side is an integer with exactly 3 factors. How many different triangles can Ian draw?

Discover how to determine the number of triangles Ian can draw with side lengths 33 and 55, where the third side has exactly 3 factors. This problem explores triangle inequalities and the properties of integers with specific factors. Perfect for high school students.

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