Solution:

Given that P × Q = 1696 and the least common multiple (LCM) of P and Q is 21879. We are also given that P and Q are composite numbers and P is a prime number. Let's solve this step by step.

Step 1: Prime Factorization of 1696

To find the prime factorization of 1696, we can start by dividing it by the smallest prime number, which is 2.

1696 ÷ 2 = 848

Now, we continue dividing 848 by 2:

848 ÷ 2 = 424

Continuing this process, we divide 424 by 2 again:

424 ÷ 2 = 212

212 ÷ 2 = 106

Since 106 is an even number, we can divide it by 2 again:

106 ÷ 2 = 53

Now, we have reached a prime number. Therefore, the prime factorization of 1696 is:

1696 = 2 × 2 × 2 × 2 × 53
1696 = 2^4 × 53

Step 2: Finding P and Q

Since P is a prime number, it can only have two factors: 1 and itself. Therefore, the prime factorization of P will be in the form of P = P × 1.

Looking at the prime factorization of 1696, we can see that P must be one of the prime factors. The other factor, Q, will be the remaining factors of 1696. So, we have:

P = 2^4 = 16 (since P is a prime number)
Q = 53

Therefore, P = 16 and Q = 53.

Step 3: Verifying the LCM

Now, let's verify if the LCM of P and Q is indeed 21879.

The LCM of two numbers is the smallest multiple that is divisible by both numbers. In this case, we are given that the LCM of P and Q is 21879.

We can calculate the LCM of 16 and 53 using the prime factorization method:

16 = 2^4
53 = 53

To find the LCM, we take the highest power of each prime factor:

LCM = 2^4 × 53 = 16 × 53 = 848

But this is not equal to 21879. Therefore, there is an error in the given information.

Conclusion:

The given information is incorrect because the LCM of P and Q is not 21879. We cannot find the values of P and Q based on the given information.