Understanding the Problem: Prime Numbers and LCM

The problem asks us to find the value of an expression involving two prime numbers, A and B. We are given two conditions:

• A and B are prime numbers.
• A is greater than B (\( A > B \)).
• The Least Common Multiple (LCM) of A and B is 209 (\( \text{LCM}(A, B) = 209 \)).

We need to calculate the value of \( B^2 - A \).

Key Property of LCM for Prime Numbers

A crucial property to remember is that for any two distinct prime numbers, their LCM is simply their product. Since A and B are prime numbers and \( A > B \), they must be distinct. Therefore, the LCM of A and B is \( A \times B \).

Given \( \text{LCM}(A, B) = 209 \), we can write the equation:

\( A \times B = 209 \)

Finding the Prime Numbers A and B

Since A and B are prime numbers and their product is 209, A and B must be the prime factors of 209. We need to find the prime factorization of 209.

Let's find the factors of 209:

• Start dividing 209 by small prime numbers (2, 3, 5, 7, 11, ...).
• 209 is not divisible by 2 (it's odd).
• The sum of digits \( 2+0+9=11 \), which is not divisible by 3, so 209 is not divisible by 3.
• 209 does not end in 0 or 5, so it's not divisible by 5.
• \( 209 \div 7 = 29 \) with a remainder, so not divisible by 7.
• \( 209 \div 11 \). Let's perform the division:

      19
     ___
  11|209
     -11
     ---
      99
     -99
     ---
       0
          

• So, \( 209 = 11 \times 19 \).

The factors of 209 are 1, 11, 19, and 209. The prime factors are 11 and 19.

Since A and B are prime numbers and their product is 209, A and B must be 11 and 19.

We are given the condition \( A > B \). Comparing the two prime factors:

• If A = 19 and B = 11, then \( 19 > 11 \), which satisfies the condition.
• If A = 11 and B = 19, then \( 11 > 19 \), which does not satisfy the condition.

Therefore, the correct values for A and B are \( A = 19 \) and \( B = 11 \).

Let's verify: A=19 (prime), B=11 (prime), \( A > B \) (\( 19 > 11 \)), \( \text{LCM}(19, 11) = 19 \times 11 = 209 \). All conditions are met.

Calculating the Value of \( B^2 - A \)

Now we need to calculate the value of the expression \( B^2 - A \) using the values we found for A and B.

Substitute \( A = 19 \) and \( B = 11 \) into the expression:

\( B^2 - A = (11)^2 - 19 \)

Calculate \( 11^2 \):

\( 11^2 = 11 \times 11 = 121 \)

Now substitute this value back into the expression:

\( 121 - 19 \)

Perform the subtraction:

\( 121 - 19 = 102 \)

Conclusion

The value of \( B^2 - A \) is 102.

Revision Table: Key Concepts Reviewed

Additional Information: Exploring Prime Numbers and Factorization

Understanding prime numbers is fundamental in number theory. Prime factorization is a unique way to break down any composite number into its prime building blocks. This is often called the Fundamental Theorem of Arithmetic. In this problem, knowing that A and B were prime numbers was the key to directly using their product for the LCM. If A and B were not prime, finding their LCM would involve a different process, typically using their prime factorizations and taking the highest power of each common prime factor.

For example, the LCM of 12 and 18:

• Prime factorization of 12: \( 2^2 \times 3^1 \)
• Prime factorization of 18: \( 2^1 \times 3^2 \)
• LCM(12, 18) = \( 2^{\text{max}(2,1)} \times 3^{\text{max}(1,2)} = 2^2 \times 3^2 = 4 \times 9 = 36 \).

However, since A and B in our problem are prime, their only factors are 1 and themselves. When finding the LCM of two distinct primes, say \( p \) and \( q \), there are no common prime factors other than 1. Thus, the LCM is simply \( p \times q \). This property significantly simplified our approach to finding A and B from their LCM.