Find the smallest number by which each number should be divided get a perfect square. In each case, find the number whose square is the new number.  a) 35972

To solve this problem, we need to determine the smallest number by which 35,972 should be divided to make it a perfect square. Then, we will identify the number whose square is the resulting perfect square.

### Step 1: Prime Factorization of 35,972

First, we perform the prime factorization of 35,972 to find its prime factors.

\[ 35972 \div 2 = 17986 \]
\[ 17986 \div 2 = 8993 \]

8993 is not divisible by 2, so we check for the next prime number, 3. 8993 is not divisible by 3 either. Then, we check for divisibility by 7.

\[ 8993 \div 7 = 1285 \]

1285 is divisible by 5:

\[ 1285 \div 5 = 257 \]

257 is a prime number.

So, the prime factorization of 35,972 is:

\[ 35972 = 2^2 \times 7 \times 5 \times 257 \]

### Step 2: Pairing the Prime Factors

For a number to be a perfect square, all the prime factors must appear in pairs. In this case:

- \( 2^2 \) is already a pair.
- 7, 5, and 257 are unpaired.

To make the number a perfect square, we would need to divide it by \( 7 \times 5 \times 257 = 8995 \).

### Step 3: Finding the New Perfect Square

When 35,972 is divided by 8,995, we get:

\[ \frac{35972}{8995} = 4 \]

The number 4 is a perfect square, and the number whose square is 4 is \( 2 \).

### Final Answer

- The smallest number by which 35,972 should be divided to obtain a perfect square is **8,995**.
- The new number, after division, is **4**.
- The number whose square is the new number is **2**.

Would you like more details on any step, or do you have any questions? Here are five related questions you might find interesting:

1. How would you find the prime factorization of a larger number like 100,000?
2. What is the method to check if a number is a perfect square without factorization?
3. How would you determine the smallest number to multiply a number to make it a perfect square?
4. Can you explain the concept of square roots and perfect squares in more depth?
5. How does the prime factorization method help in identifying perfect cubes?

**Tip:** When performing prime factorization, always start with the smallest prime numbers and work your way up to ensure accuracy.

Learn how to find the smallest number to make 35972 a perfect square with detailed step-by-step solution. This problem covers prime factorization, perfect squares, and divisibility rules. Suitable for students in Grades 7-9.

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