Understanding the Prime Number Problem
The question asks us to find the number of possible combinations for \((x, y, x+y, x - y)\) given specific conditions. Let's break down these conditions:
- \(x\) and \(y\) must be natural numbers. Natural numbers are positive whole numbers: \(\{1, 2, 3, ...\}\).
- Both \(x\) and \(y\) must be less than 20. So, \(1 \le x \le 19\) and \(1 \le y \le 19\).
- The numbers \(x\), \(y\), \(x+y\), and \(x-y\) must all be prime numbers.
First, let's list the prime numbers less than 20:
Primes < 20 = \(\{2, 3, 5, 7, 11, 13, 17, 19\}\)
Since \(x-y\) must be a prime number, it must be positive. This means \(x\) must be greater than \(y\) (\(x > y\)).
Analyzing the Conditions for x and y
We need \(x\) and \(y\) to be prime numbers themselves, and \(x > y\). The possible pairs of \((x, y)\) where both are primes less than 20 and \(x > y\) are:
- \(y=2\), \(x \in \{3, 5, 7, 11, 13, 17, 19\}\)
- \(y=3\), \(x \in \{5, 7, 11, 13, 17, 19\}\)
- \(y=5\), \(x \in \{7, 11, 13, 17, 19\}\)
- \(y=7\), \(x \in \{11, 13, 17, 19\}\)
- \(y=11\), \(x \in \{13, 17, 19\}\)
- \(y=13\), \(x \in \{17, 19\}\)
- \(y=17\), \(x \in \{19\}\)
Now let's consider the parity (even or odd) of \(x\) and \(y\). Remember, the only even prime number is 2.
- Case 1: \(y\) is an odd prime.
If \(y\) is an odd prime (like 3, 5, 7, etc.), then \(x\) must also be an odd prime (since \(x > y\) and \(x\) must be prime).- If both \(x\) and \(y\) are odd primes, then \(x-y\) (odd - odd) would be an even number. The only even prime is 2. So, \(x-y = 2\).
- Similarly, \(x+y\) (odd + odd) would be an even number. For \(x+y\) to be prime, it must be 2. However, since \(x > y \ge 3\), \(x+y\) must be greater than \(3+3=6\). Thus, \(x+y\) cannot be 2.
- Therefore, it's impossible for both \(x\) and \(y\) to be odd primes.
- Case 2: \(y\) is the even prime.
The only possibility is \(y=2\). Since \(x > y\), \(x\) must be an odd prime.- We need \(x\) (odd prime < 20), \(y=2\) (prime), \(x+y\) (prime), and \(x-y\) (prime).
- Let's check the possible values for \(x\) from the list of odd primes greater than 2 and less than 20: \(x \in \{3, 5, 7, 11, 13, 17, 19\}\).
Testing Potential Combinations
We will test each possible value of \(x\) (where \(y=2\)):
- If \(x=3, y=2\):
- \(x=3\) (prime), \(y=2\) (prime)
- \(x+y = 3+2 = 5\) (prime)
- \(x-y = 3-2 = 1\) (not prime)
- If \(x=5, y=2\):
- \(x=5\) (prime), \(y=2\) (prime)
- \(x+y = 5+2 = 7\) (prime)
- \(x-y = 5-2 = 3\) (prime)
- If \(x=7, y=2\):
- \(x=7\) (prime), \(y=2\) (prime)
- \(x+y = 7+2 = 9\) (not prime)
- \(x-y = 7-2 = 5\) (prime)
- If \(x=11, y=2\):
- \(x=11\) (prime), \(y=2\) (prime)
- \(x+y = 11+2 = 13\) (prime)
- \(x-y = 11-2 = 9\) (not prime)
- If \(x=13, y=2\):
- \(x=13\) (prime), \(y=2\) (prime)
- \(x+y = 13+2 = 15\) (not prime)
- \(x-y = 13-2 = 11\) (prime)
- If \(x=17, y=2\):
- \(x=17\) (prime), \(y=2\) (prime)
- \(x+y = 17+2 = 19\) (prime)
- \(x-y = 17-2 = 15\) (not prime)
- If \(x=19, y=2\):
- \(x=19\) (prime), \(y=2\) (prime)
- \(x+y = 19+2 = 21\) (not prime)
- \(x-y = 19-2 = 17\) (prime)
Conclusion on Combinations
By systematically checking all possibilities based on the properties of prime numbers and the given conditions, we found only one pair \((x, y)\) that satisfies all requirements:
- \((x, y) = (5, 2)\)
This leads to the combination \((x, y, x+y, x-y) = (5, 2, 7, 3)\). All numbers in this tuple are prime, and \(x=5\) and \(y=2\) are natural numbers less than 20.
Therefore, there is only one such combination possible.
Similar Questions
- Let \(p\) and \(q\) be natural numbers such that \(q > p\). What is the largest value of \(p\) such that \(q^2 - 5p-4\) is negative?
Let \(x = n(n+1)(n+2)\), where \(n\) is an even natural number. Which of the following statements is/are correct?
I. \(x\) is always divisible by 48.
II. \(x^2\) is always divisible by 144.
Select the answer using the code given below.
- A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Is \(r^n\) less than 1, where \(r\) is a real number and \(n\) is a natural number ?
Statement-I : \(0 < r^2 < 1\)
Statement-II : \(0 < r^3 < 2\)
Which one of the following is correct in respect of the above Question and the Statements ? - A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Is \((0.5)^n + (0.5)^{-n}\) always greater than 2, where \(n\) is an integer ?
Statement-I : \(n\) is an even integer
Statement-II : \(n\) is negative
Which one of the following is correct in respect of the above Question and the Statements ? - A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Let XYZ be a 3-digit number and the difference between XYZ and ZYX is equal to PQR. Is (P + R) equal to Q ?
Statement-I : P = 3
Statement-II : R = 6
Which one of the following is correct in respect of the above Question and the Statements ? - A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Is \((p^2 + q^2)\) always composite number, where \(p\) and \(q\) are different prime numbers ?
Statement-I : \((p - q)\) is an odd integer
Statement-II : \((p + q)\) is an odd integer
Which one of the following is correct in respect of the above Question and the Statements ? What is the remainder when
\((17^{25} +19^{25})\)
is divided by 18?
- If \(n\) is natural number less than 7, then what is the number of values of \(n\) for which \((12n+2)\) and \((8n+1)\) are relatively prime?
- Let XYZ be a 3-digit number. Let D be the difference between XYZ and ZYX. What is the remainder when D is divided by 99?
- Let \(p\) and \(q\) be two natural numbers such that \((p+q)^{p+q}\) is divisible by 512. What is the least value of \((p+q)\)?
Important Questions from Number System
What is the value of 1 2 + 2 2 + 3 2 + ......21 2 ?
Which sequence is correct to represent the hierarchical chain of number system?
(Where N - Natural Numbers
W - Whole Numbers
Q - Rational Numbers
Z - Integers)
The difference of the place value and the face value of 5 in 26549 is :
What must be added to 45680 to make it exactly divisible by 9?
How many zeroes are there at the end of the following product?
1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60
