Finding the Least Divisor for a Perfect Square

The problem asks us to find the smallest number by which 1568 should be divided to obtain a perfect square. A perfect square is a number that can be expressed as the product of an integer by itself (e.g., 9 is a perfect square because $3 \times 3 = 9$). To solve this, we need to understand the prime factorization of the number 1568.

Prime Factorization of 1568

First, let's find the prime factors of 1568. We do this by repeatedly dividing the number by the smallest possible prime numbers:

• Start with 1568. Since it's an even number, divide by 2: $1568 \div 2 = 784$
• 784 is also even, divide by 2: $784 \div 2 = 392$
• 392 is even, divide by 2: $392 \div 2 = 196$
• 196 is even, divide by 2: $196 \div 2 = 98$
• 98 is even, divide by 2: $98 \div 2 = 49$
• 49 is not divisible by 2, 3, or 5. The next prime number is 7. Divide by 7: $49 \div 7 = 7$
• Finally, divide by 7: $7 \div 7 = 1$

So, the prime factorization of 1568 is $2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7$, which can be written using exponents as $2^5 \times 7^2$.

Making the Number a Perfect Square

For a number to be a perfect square, all the exponents in its prime factorization must be even numbers. Let's look at the exponents in the prime factorization of 1568 ($2^5 \times 7^2$):

• The prime factor 2 has an exponent of 5, which is odd.
• The prime factor 7 has an exponent of 2, which is even.

To make the number a perfect square, we need to adjust the prime factors so that all exponents become even. We can achieve this by dividing the original number (1568) by the prime factors that have odd exponents, specifically by raising those factors to the power needed to make the exponent even. In this case, the exponent of 2 is 5. To make it even, we need to reduce it to the nearest lower even number, which is 4. We can do this by dividing by $2^{(5-4)} = 2^1 = 2$.

The prime factor 7 already has an even exponent (2), so we don't need to do anything with it.

Determining the Least Divisor

The least number we should divide 1568 by is the factor that removes the 'odd exponent' part from the prime factorization. This factor is $2^1$, which is simply 2.

Therefore, the least number by which 1568 should be divided is 2.

Verification

Let's divide 1568 by 2 and check if the result is a perfect square:

Now, let's find the prime factorization of 784:

In the prime factorization of 784 ($2^4 \times 7^2$), both exponents (4 and 2) are even. This confirms that 784 is a perfect square. We can also see that $784 = (2^2 \times 7^1)^2 = (4 \times 7)^2 = 28^2$.

Thus, dividing 1568 by the least number 2 results in a perfect square (784).