A perfect square is a positive integer that is the result of squaring a positive integer. If `N = 3^4 cdot 5^3 cdot 7,` then what is the biggest perfect square that is a factor of N?

Options

A. `3^2`
B. `5^2`
C. `9^2`
D. `(9 cdot 5)^(2)`

Answer

To find the biggest perfect square that is a factor of \( N = 3^4 \cdot 5^3 \cdot 7 \), we will follow these steps:

### Step 1: Identify the prime factorization of \( N \)
We have:
\[ N = 3^4 \cdot 5^3 \cdot 7^1 \]

### Step 2: Determine the highest even powers of each prime factor
A perfect square must have even powers for all its prime factors. We will analyze the powers of each prime factor in \( N \):

- For \( 3^4 \): The highest even power is \( 4 \) (which is \( 3^4 \)).
- For \( 5^3 \): The highest even power is \( 2 \) (which is \( 5^2 \)).
- For \( 7^1 \): The highest even power is \( 0 \) (which is \( 7^0 = 1 \)) since \( 1 \) is not included in the perfect square.

### Step 3: Construct the perfect square
Now we can construct the biggest perfect square factor of \( N \) using the highest even powers we found:

\[ \text{Perfect square} = 3^4 \cdot 5^2 \cdot 7^0 = 3^4 \cdot 5^2 \]

### Step 4: Simplify the expression
We can express this as:
\[ 3^4 = (3^2)^2 = 9^2 \]
\[ 5^2 = (5)^2 \]
Thus, we can write:
\[ \text{Perfect square} = (3^2 \cdot 5)^2 = (9 \cdot 5)^2 = 45^2 \]

### Step 5: Final answer
The biggest perfect square that is a factor of \( N \) is:
\[ 9 \cdot 5 = 45 \]
So, the biggest perfect square factor of \( N \) is \( 9 \cdot 5 \) whole square.

### Conclusion
The biggest perfect square that is a factor of \( N \) is \( (9 \cdot 5)^2 \) or \( 45^2 \).

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