To find the least number that should be added to the product \( 11 \times 12 \times 13 \times 14 \) to make it a perfect square, we can follow these steps:
### Step 1: Calculate the product \( 11 \times 12 \times 13 \times 14 \)
First, we need to calculate the product of these four numbers.
\[
11 \times 12 = 132
\]
\[
132 \times 13 = 1716
\]
\[
1716 \times 14 = 24024
\]
So, the product \( 11 \times 12 \times 13 \times 14 = 24024 \).
### Step 2: Determine if \( 24024 \) is a perfect square
Next, we need to check if \( 24024 \) is a perfect square. A number is a perfect square if all the prime factors in its prime factorization have even exponents.
### Step 3: Find the prime factorization of \( 24024 \)
To find the prime factorization, we can divide \( 24024 \) by prime numbers:
\[
24024 \div 2 = 12012
\]
\[
12012 \div 2 = 6006
\]
\[
6006 \div 2 = 3003
\]
\[
3003 \div 3 = 1001
\]
\[
1001 \div 7 = 143
\]
\[
143 \div 11 = 13
\]
\[
13 \div 13 = 1
\]
Thus, the prime factorization of \( 24024 \) is:
\[
24024 = 2^3 \times 3^1 \times 7^1 \times 11^1 \times 13^1
\]
### Step 4: Analyze the exponents
Now, we check the exponents of the prime factors:
- \( 2^3 \) (exponent is 3, which is odd)
- \( 3^1 \) (exponent is 1, which is odd)
- \( 7^1 \) (exponent is 1, which is odd)
- \( 11^1 \) (exponent is 1, which is odd)
- \( 13^1 \) (exponent is 1, which is odd)
Since all the prime factors have odd exponents, we need to add enough to make all exponents even.
### Step 5: Determine the least number to add
To make the exponents even, we can add the following:
- For \( 2^3 \), we need to add \( 1 \) (to make it \( 2^4 \)).
- For \( 3^1 \), we need to add \( 2 \) (to make it \( 3^2 \)).
- For \( 7^1 \), we need to add \( 2 \) (to make it \( 7^2 \)).
- For \( 11^1 \), we need to add \( 2 \) (to make it \( 11^2 \)).
- For \( 13^1 \), we need to add \( 2 \) (to make it \( 13^2 \)).
The least number that should be added to \( 24024 \) to make it a perfect square is \( 1 \).
### Conclusion
Thus, the least number that should be added to \( 24024 \) to make it a perfect square is \( 1 \).
### Final Answer
The answer is \( 1 \).
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