Determining Relative Primality for Given Expressions
The problem asks us to find the number of possible values for a natural number \(n\), where \(n\) is less than 7, such that the two expressions, \(12n+2\) and \(8n+1\), are relatively prime.
Understanding Relative Primality
Two integers are considered relatively prime (or coprime) if their greatest common divisor (GCD) is equal to 1. Mathematically, integers \(a\) and \(b\) are relatively prime if \(\text{gcd}(a, b) = 1\).
Identifying the Range of 'n'
The question specifies that \(n\) is a natural number and \(n < 7\). Natural numbers typically start from 1. Therefore, the possible values for \(n\) are:
- \(n \in \{1, 2, 3, 4, 5, 6\}\)
We need to check for which of these values of \(n\) the expressions \(12n+2\) and \(8n+1\) are relatively prime.
Calculating the Greatest Common Divisor (GCD)
Let's find the GCD of the two expressions, \(12n+2\) and \(8n+1\). We can use the properties of GCD. Let \(d = \text{gcd}(12n+2, 8n+1)\).
According to the properties of GCD, \(d\) must divide any integer linear combination of \(12n+2\) and \(8n+1\). We can manipulate these expressions to eliminate \(n\).
Consider the multiples of the expressions:
- Multiply \((8n+1)\) by 2: \(2 \times (8n+1) = 16n+2\).
- Multiply \((12n+2)\) by 1: \(1 \times (12n+2) = 12n+2\).
Alternatively, we can use the property \(\text{gcd}(a, b) = \text{gcd}(a, b-ka)\). Let \(a = 12n+2\) and \(b = 8n+1\). We aim to simplify the expression.
Using the Euclidean algorithm property, \(\text{gcd}(a, b) = \text{gcd}(a - k \cdot b, b)\). Let's try to eliminate the highest power term (\(n\)).
We can write:
\(\text{gcd}(12n+2, 8n+1)\)
\(= \text{gcd}(12n+2 - 1 \cdot (8n+1), 8n+1)\)
\(= \text{gcd}(12n+2 - 8n - 1, 8n+1)\)
\(= \text{gcd}(4n+1, 8n+1)\)
Now, apply the property again:
\(\text{gcd}(4n+1, 8n+1) = \text{gcd}(4n+1, 8n+1 - 2 \cdot (4n+1))\)
\(= \text{gcd}(4n+1, 8n+1 - 8n - 2)\)
\(= \text{gcd}(4n+1, -1)\)
The greatest common divisor of any integer \(x\) and \(-1\) is always 1 (since GCD is defined as a positive integer). Therefore:
\(\text{gcd}(4n+1, -1) = 1\).
This calculation shows that \(\text{gcd}(12n+2, 8n+1) = 1\) for all integer values of \(n\). This means the expressions \(12n+2\) and \(8n+1\) are always relatively prime, regardless of the value of \(n\).
Counting the Values of 'n'
Since the expressions are relatively prime for all integer values of \(n\), they will certainly be relatively prime for all the allowed values of \(n\) in the set \(\{1, 2, 3, 4, 5, 6\}\).
The number of possible values for \(n\) is the count of elements in this set, which is 6.
Conclusion
There are 6 values of \(n\) (namely 1, 2, 3, 4, 5, and 6) for which \(n\) is a natural number less than 7 and \((12n+2)\) and \((8n+1)\) are relatively prime.
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 \(x\) 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.
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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.
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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.
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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?
- 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?
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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
