The LCM of two prime numbers p and q is 2231, where p > q. What is the value of p - q ?
74
Understanding the Problem: LCM of Prime Numbers
The question asks us to find the difference between two prime numbers, p and q, given that their Least Common Multiple (LCM) is 2231 and p is greater than q. We need to use our knowledge of prime numbers and LCM to solve this problem.
Key Concept: LCM of Two Prime Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples are 2, 3, 5, 7, 11, etc.
The LCM of two numbers is the smallest positive integer that is a multiple of both numbers.
For two distinct prime numbers, say p and q, their only common positive divisor is 1. Therefore, their Least Common Multiple (LCM) is simply their product.
Mathematically, for prime numbers p and q:
- GCD(p, q) = 1 (Greatest Common Divisor)
- LCM(p, q) = p × q
This key property is crucial for solving the problem.
Solving the Problem: Finding the Prime Numbers
We are given that the LCM of the two prime numbers p and q is 2231. Using the property discussed above, we know that:
\( p \times q = 2231 \)
Since p and q are prime numbers, finding p and q is equivalent to finding the prime factors of 2231.
We need to find two prime numbers whose product is 2231. Let's find the prime factors of 2231 by testing divisibility by small prime numbers:
- Is 2231 divisible by 2? No, it's an odd number.
- Is 2231 divisible by 3? The sum of digits is 2+2+3+1 = 8, which is not divisible by 3. So, 2231 is not divisible by 3.
- Is 2231 divisible by 5? No, it doesn't end in 0 or 5.
- Is 2231 divisible by 7? \( 2231 \div 7 \approx 318.7 \). No.
- Is 2231 divisible by 11? \( 2231 = 11 \times 202 + 9 \). No. (Alternating sum of digits: 1-3+2-2 = -2, not divisible by 11).
- Is 2231 divisible by 13? \( 2231 = 13 \times 171 + 8 \). No.
- Is 2231 divisible by 17? \( 2231 = 17 \times 131 + 4 \). No.
- Is 2231 divisible by 19? \( 2231 = 19 \times 117 + 8 \). No.
- Is 2231 divisible by 23? Let's try: \( 2231 \div 23 \). \( 23 \times 100 = 2300 \) \( 23 \times 10 = 230 \) \( 23 \times 3 = 69 \) \( 23 \times 90 = 2070 \) \( 2231 - 2070 = 161 \) \( 23 \times 7 = 161 \) So, \( 2231 = 23 \times (90 + 7) = 23 \times 97 \).
We found that 2231 can be expressed as the product of 23 and 97. Now we need to check if both 23 and 97 are prime numbers.
- 23 is a prime number (only divisible by 1 and 23).
- To check if 97 is prime, we can test divisibility by primes up to the square root of 97, which is approximately 9.8. The primes to test are 2, 3, 5, 7.
- 97 is not divisible by 2 (odd).
- 9+7 = 16, not divisible by 3.
- Doesn't end in 0 or 5, not divisible by 5.
- \( 97 \div 7 = 13 \) with a remainder of 6. Not divisible by 7.
Since 97 is not divisible by any prime number less than or equal to its square root (other than 1), 97 is a prime number.
So, the two prime factors of 2231 are 23 and 97.
Determining p and q and Calculating the Difference
We found that \( p \times q = 97 \times 23 \). The problem states that \( p > q \). Therefore, we must have:
- \( p = 97 \)
- \( q = 23 \)
Both 97 and 23 are prime numbers, and 97 is indeed greater than 23.
The question asks for the value of \( p - q \).
\( p - q = 97 - 23 \)
Let's calculate the difference:
\( 97 - 23 = 74 \)
The value of \( p - q \) is 74.
Conclusion
Given that p and q are prime numbers with LCM(p, q) = 2231 and p > q, we found that the prime factors of 2231 are 97 and 23. Assigning p = 97 and q = 23 satisfies the condition p > q. The difference p - q is 97 - 23 = 74.
Revision Table: Key Facts
| Concept | Description | Application in Problem |
|---|---|---|
| Prime Number | A number > 1 with only two divisors: 1 and itself. | p and q are prime numbers. |
| LCM (Least Common Multiple) | Smallest positive multiple common to two numbers. | Given LCM(p, q) = 2231. |
| LCM of Two Primes | For distinct primes p and q, LCM(p, q) = p × q. | \( p \times q = 2231 \). |
| Prime Factorization | Expressing a number as a product of its prime factors. | Used to find p and q from 2231. |
Additional Information: Properties of Prime Numbers and LCM
- Any integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic). This is why finding the prime factors of 2231 gives us the specific prime numbers p and q.
- The relationship between LCM and GCD of two positive integers a and b is: \( \text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b \).
- For two distinct prime numbers p and q, GCD(p, q) = 1. Using the formula: \( \text{LCM}(p, q) \times 1 = p \times q \), which confirms that LCM(p, q) = p × q for distinct primes.
- Prime numbers are the building blocks of all integers through multiplication.
- Knowing how to find prime factors is essential for many number theory problems, including those involving LCM and GCD.
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