Finding Prime Numbers A and B and Calculating the Expression

The problem states that the product of two prime numbers, A and B, is 221, and A is less than B (A < B). We need to find the value of the expression \(4A - 3B\).

What are Prime Numbers?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, 13, 17, 19, and so on.

Finding the Prime Factors of 221

Since A and B are prime numbers and their product \(A \times B = 221\), A and B must be the prime factors of 221. To find these factors, we can test for divisibility by small prime numbers.

• Is 221 divisible by 2? No (it's an odd number).
• Is 221 divisible by 3? The sum of digits is 2+2+1 = 5, which is not divisible by 3. So, 221 is not divisible by 3.
• Is 221 divisible by 5? No (it doesn't end in 0 or 5).
• Is 221 divisible by 7? \(221 \div 7 \approx 31.57\). No.
• Is 221 divisible by 11? \(221 = 11 \times 20 + 1\). No.
• Is 221 divisible by 13? Let's try \(221 \div 13\):

  \(13 \times 10 = 130\)

  \(221 - 130 = 91\)

  \(13 \times 7 = 91\)

  So, \(221 = 130 + 91 = 13 \times 10 + 13 \times 7 = 13 \times (10 + 7) = 13 \times 17\).

  
  Yes, 221 is divisible by 13, and the result is 17.

Both 13 and 17 are prime numbers.

Identifying A and B

We found that the prime factors of 221 are 13 and 17. The problem states that A < B. Therefore, we assign the smaller prime factor to A and the larger prime factor to B.

• A = 13
• B = 17

We can verify this: A and B are prime numbers, A < B (13 < 17), and their product \(A \times B = 13 \times 17 = 221\).

Calculating the Value of (4A – 3B)

Now we substitute the values of A and B into the expression \(4A - 3B\):

\(4A - 3B = 4 \times (13) - 3 \times (17)\)

First, calculate the products:

• \(4 \times 13 = 52\)
• \(3 \times 17 = 51\)

Now, perform the subtraction:

\(52 - 51 = 1\)

So, the value of the expression \(4A - 3B\) is 1.

Comparing with Options

Let's compare our result with the given options:

• Option 1: -1
• Option 2: 2
• Option 3: 1
• Option 4: -2

Our calculated value is 1, which matches Option 3.

Revision Table: Prime Numbers and Factors

Additional Information on Finding Prime Factors

To find the prime factors of a number, especially a smaller number like 221, you can systematically test for divisibility by prime numbers starting from the smallest (2, 3, 5, 7, 11, 13, etc.). You only need to test prime divisors up to the square root of the number you are factoring. The square root of 221 is approximately 14.8. So, we only needed to check prime numbers up to 13.

• Start with the smallest prime number, 2. If the number is divisible, divide it and repeat with the quotient.
• If not divisible by 2, try the next prime number, 3. Repeat if divisible.
• Continue this process with prime numbers 5, 7, 11, 13, and so on, until you reach a prime factor or the remaining number is prime.

In the case of 221, we found it was not divisible by 2, 3, 5, 7, or 11. When we tried 13, we found that \(221 \div 13 = 17\). Since both 13 and 17 are prime numbers, these are the prime factors of 221.