By what least number should the given number me multiplied to get a perfect square number? In each case, find the number whose square is the new number.
3675
2156
3332
1925
9075
7623
3380
2475
1575
9075
4851
3380
4500
7776
8820
4056

Answer

To solve the problem of finding the least number by which a given number should be multiplied to obtain a perfect square, we will follow these steps for each number provided.

### Step-by-Step Solution:

1. **Prime Factorization**: Start by performing the prime factorization of the given number.
2. **Identify Pairs**: Count the occurrences of each prime factor. For a number to be a perfect square, all prime factors must occur in pairs.
3. **Determine the Missing Factors**: Identify which prime factors have an odd count. Each of these factors needs to be multiplied to make their count even.
4. **Calculate the Least Number**: Multiply the identified missing factors to get the least number that should be multiplied.
5. **Calculate the New Perfect Square**: Multiply the original number by the least number found in the previous step to get the new number, and then find the square root of this new number.

Let's apply these steps to each of the given numbers.

### Example Calculations:

1. **3675**
- Prime Factorization: \(3675 = 3^1 \times 5^2 \times 7^1\)
- Odd Count Factors: \(3^1\) and \(7^1\) (both have odd counts).
- Least Number to Multiply: \(3 \times 7 = 21\)
- New Number: \(3675 \times 21 = 77175\)
- Square Root of New Number: \(\sqrt{77175} = 105\)

2. **2156**
- Prime Factorization: \(2156 = 2^2 \times 7^2 \times 11^1\)
- Odd Count Factors: \(11^1\)
- Least Number to Multiply: \(11\)
- New Number: \(2156 \times 11 = 23716\)
- Square Root of New Number: \(\sqrt{23716} = 154\)

3. **3332**
- Prime Factorization: \(3332 = 2^2 \times 7^2 \times 17^1\)
- Odd Count Factors: \(17^1\)
- Least Number to Multiply: \(17\)
- New Number: \(3332 \times 17 = 56644\)
- Square Root of New Number: \(\sqrt{56644} = 238\)

4. **1925**
- Prime Factorization: \(1925 = 5^2 \times 7^1 \times 11^1\)
- Odd Count Factors: \(7^1\) and \(11^1\)
- Least Number to Multiply: \(7 \times 11 = 77\)
- New Number: \(1925 \times 77 = 148225\)
- Square Root of New Number: \(\sqrt{148225} = 385\)

5. **9075**
- Prime Factorization: \(9075 = 3^2 \times 5^2 \times 11^1\)
- Odd Count Factors: \(11^1\)
- Least Number to Multiply: \(11\)
- New Number: \(9075 \times 11 = 99825\)
- Square Root of New Number: \(\sqrt{99825} = 315\)

6. **7623**
- Prime Factorization: \(7623 = 3^2 \times 7^1 \times 11^1\)
- Odd Count Factors: \(7^1\) and \(11^1\)
- Least Number to Multiply: \(7 \times 11 = 77\)
- New Number: \(7623 \times 77 = 587769\)
- Square Root of New Number: \(\sqrt{587769} = 765\)

7. **3380**
- Prime Factorization: \(3380 = 2^2 \times 5^1 \times 13^1\)
- Odd Count Factors: \(5^1\) and \(13^1\)
- Least Number to Multiply: \(5 \times 13 = 65\)
- New Number: \(3380 \times 65 = 219700\)
- Square Root of New Number: \(\sqrt{219700} = 469\)

8. **2475**
- Prime Factorization: \(2475 = 3^1 \times 5^2 \times 11^1\)
- Odd Count Factors: \(3^1\) and \(11^1\)
- Least Number to Multiply: \(3 \times 11 = 33\)
- New Number: \(2475 \times 33 = 81675\)
- Square Root of New Number: \(\sqrt{81675} = 285\)

9. **1575**
- Prime Factorization: \(1575 = 3^2 \times 5^2 \times 7^1\)
- Odd Count Factors: \(7^1\)
- Least Number to Multiply: \(7\)
- New Number: \(1575 \times 7 = 11025\)
- Square Root of New Number: \(\sqrt{11025} = 105\)

10. **4851**
- Prime Factorization: \(4851 = 3^1 \times 7^1 \times 231\) (continue factoring)
- Odd Count Factors: \(3^1\) and \(7^1\)
- Least Number to Multiply: \(3 \times 7 = 21\)
- New Number: \(4851 \times 21 = 101871\)
- Square Root of New Number: \(\sqrt{101871} = 319\)

11. **4500**
- Prime Factorization: \(4500 = 2^2 \times 3^2 \times 5^3\)
- Odd Count Factors: \(5^1\)
- Least Number to Multiply: \(5\)
- New Number: \(4500 \times 5 = 22500\)
- Square Root of New Number: \(\sqrt{22500} = 150\)

12. **7776**
- Prime Factorization: \(7776 = 2^5 \times 3^5\)
- All counts are even, so no multiplication is needed.
- New Number: \(7776\)
- Square Root of New Number: \(\sqrt{7776} = 88\)

13. **8820**
- Prime Factorization: \(8820 = 2^2 \times 3^2 \times 5^1 \times 7^1\)
- Odd Count Factors: \(5^1\) and \(7^1\)
- Least Number to Multiply: \(5 \times 7 = 35\)
- New Number: \(8820 \times 35 = 308700\)
- Square Root of New Number: \(\sqrt{308700} = 555\)

14. **4056**
- Prime Factorization: \(4056 = 2^3 \times 3^2 \times 7^1\)
- Odd Count Factors: \(7^1\)
- Least Number to Multiply: \(7\)
- New Number: \(4056 \times 7 = 28392\)
- Square Root of New Number: \(\sqrt{28392} = 168\)