Given that 4704/m = n ^ 2, where m and n are whole numbers and n is as large as possible, find value of m and n.

To solve the problem where \( \frac{4704}{m} = n^2 \) and \( m \) and \( n \) are whole numbers, with \( n \) being as large as possible, we proceed as follows:

### Step 1: Factorize 4704
First, we need to factorize 4704 into its prime factors:

\[
4704 \div 2 = 2352
\]
\[
2352 \div 2 = 1176
\]
\[
1176 \div 2 = 588
\]
\[
588 \div 2 = 294
\]
\[
294 \div 2 = 147
\]
\[
147 \div 3 = 49
\]
\[
49 \div 7 = 7
\]
\[
7 \div 7 = 1
\]

Thus, the prime factorization of 4704 is:

\[
4704 = 2^5 \times 3 \times 7^2
\]

### Step 2: Find \( m \) such that \( \frac{4704}{m} = n^2 \) is a perfect square
For \( n^2 \) to be a perfect square, the powers of all prime factors in \( n^2 \) must be even. This implies that the factor \( m \) must cancel out any prime factor in 4704 that has an odd exponent.

In our factorization:
- \( 2^5 \) has an odd exponent (5).
- \( 3^1 \) has an odd exponent (1).
- \( 7^2 \) has an even exponent (2).

To make \( n^2 \) a perfect square, we need to cancel out the odd exponents. Therefore, we should have:

\[
m = 2^1 \times 3^1 = 2 \times 3 = 6
\]

### Step 3: Calculate \( n \)
Given \( m = 6 \):

\[
n^2 = \frac{4704}{6} = \frac{4704}{6} = 784
\]

Taking the square root of 784:

\[
n = \sqrt{784} = 28
\]

### Conclusion:
The value of \( m \) is 6, and the largest possible value of \( n \) is 28.

Would you like further details or have any questions?

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**Related Questions:**
1. What is the general method to find the largest possible value of \( n \) for any given number?
2. How would the problem change if \( n \) was required to be an even number?
3. Can \( m \) be greater than 4704? Why or why not?
4. How would you approach this problem if \( n \) was required to be a prime number?
5. How can this approach be used to solve similar problems with different numbers?
6. How does the factorization of 4704 influence the solution?
7. Can there be other values of \( m \) and \( n \) that satisfy the condition?
8. What if the equation was \( \frac{4704}{m} = n^3 \) instead of \( n^2 \)?

**Tip:** Always break down a number into its prime factors when dealing with perfect squares or perfect cubes; this simplifies the problem significantly.

Discover how to find the values of m and n in the equation 4704/m = n^2 with a detailed step-by-step solution. Learn about prime factorization, perfect squares, and how to determine the largest possible value of n. Suitable for Grades 7-9.

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