Solving for the Largest Value of p
We are given a problem involving two natural numbers, \(p\) and \(q\), with the condition that \(q > p\). We need to find the largest possible value for \(p\) such that the expression \(q^2 - 5p - 4\) is negative.
Understanding the Inequality
The core condition is that \(q^2 - 5p - 4 < 0\). This is an inequality involving \(p\) and \(q\). Natural numbers are positive whole numbers, so \(p, q \in \{1, 2, 3, ...\}\).
We can rewrite the inequality to isolate \(q^2\): \(q^2 < 5p + 4\)
Analyzing the Constraints
We also know that \(q > p\). Since \(p\) and \(q\) must be natural numbers, the smallest possible value for \(q\) is \(p+1\). We need to find the largest natural number \(p\) for which there exists at least one natural number \(q\) satisfying both \(q > p\) and \(q^2 < 5p + 4\).
Step-by-Step Analysis of Potential p Values
Let's test the possible values for \(p\) provided in the options. We are looking for the largest value of \(p\) that works. The options are 3, 4, 5, and 6.
Case 1: \(p = 3\)
- Condition: \(q > 3\) and \(q^2 < 5(3) + 4\).
- Simplified inequality: \(q^2 < 15 + 4\), which means \(q^2 < 19\).
- We need a natural number \(q\) such that \(q > 3\) and \(q^2 < 19\).
- Let's check values for \(q\) starting from \(4\) (since \(q > 3\)):
- If \(q=4\), then \(q^2 = 16\). Since \(16 < 19\), this condition is met.
- Since we found a valid \(q\) (e.g., \(q=4\)) for \(p=3\), the value \(p=3\) is a possible solution.
Case 2: \(p = 4\)
- Condition: \(q > 4\) and \(q^2 < 5(4) + 4\).
- Simplified inequality: \(q^2 < 20 + 4\), which means \(q^2 < 24\).
- We need a natural number \(q\) such that \(q > 4\) and \(q^2 < 24\).
- Let's check values for \(q\) starting from \(5\) (since \(q > 4\)):
- If \(q=5\), then \(q^2 = 25\). Since \(25\) is NOT less than \(24\), this condition is not met.
- For any \(q \ge 5\), \(q^2\) will be \(\ge 25\), so \(q^2 < 24\) cannot be satisfied.
- Therefore, there is no natural number \(q\) satisfying the conditions when \(p=4\). So, \(p=4\) is not a possible value.
Case 3: \(p = 5\)
- Condition: \(q > 5\) and \(q^2 < 5(5) + 4\).
- Simplified inequality: \(q^2 < 25 + 4\), which means \(q^2 < 29\).
- We need a natural number \(q\) such that \(q > 5\) and \(q^2 < 29\).
- Let's check values for \(q\) starting from \(6\) (since \(q > 5\)):
- If \(q=6\), then \(q^2 = 36\). Since \(36\) is NOT less than \(29\), this condition is not met.
- For any \(q \ge 6\), \(q^2\) will be \(\ge 36\), so \(q^2 < 29\) cannot be satisfied.
- Therefore, there is no natural number \(q\) satisfying the conditions when \(p=5\). So, \(p=5\) is not a possible value.
Case 4: \(p = 6\)
- Condition: \(q > 6\) and \(q^2 < 5(6) + 4\).
- Simplified inequality: \(q^2 < 30 + 4\), which means \(q^2 < 34\).
- We need a natural number \(q\) such that \(q > 6\) and \(q^2 < 34\).
- Let's check values for \(q\) starting from \(7\) (since \(q > 6\)):
- If \(q=7\), then \(q^2 = 49\). Since \(49\) is NOT less than \(34\), this condition is not met.
- For any \(q \ge 7\), \(q^2\) will be \(\ge 49\), so \(q^2 < 34\) cannot be satisfied.
- Therefore, there is no natural number \(q\) satisfying the conditions when \(p=6\). So, \(p=6\) is not a possible value.
Conclusion
By testing the possible values for \(p\), we found that only \(p=3\) allows for a natural number \(q\) (\(q>p\)) such that \(q^2 - 5p - 4 < 0\). Therefore, the largest value of \(p\) that satisfies the given conditions is 3.
Similar Questions
- Let \(x\) and \(y\) be natural numbers, each less than 20, such that \(x\), \(y\), \(x+y\) and \(x-y\) are prime numbers. How many such combinations of \((x, y, x+y, x - y)\) are possible?
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
