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In linear diophantine equations of the form ax + by = c, where a, b, and c are integers, which of the following is true?
- a)A solution exists if c is a multiple of gcd(a, b).
- b)A solution exists if c is a prime number.
- c)A solution exists if a and b are prime numbers.
- d)A solution exists if c is an odd number.
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
In linear diophantine equations of the form ax + by = c, where a, b, a...
In linear diophantine equations, a solution exists if c is a multiple of the greatest common divisor (GCD) of a and b.
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In linear diophantine equations of the form ax + by = c, where a, b, a...
Linear Diophantine Equations:
A linear Diophantine equation is an equation of the form ax + by = c, where a, b, and c are integers, and x and y are variables. The goal is to find integer solutions for x and y that satisfy the equation.
Existence of Solutions:
In the given linear Diophantine equation ax + by = c, a solution exists if and only if c is a multiple of the greatest common divisor (gcd) of a and b.
Explanation:
To understand why option 'A' is true, let's consider the concept of the gcd and its relation to the solution of a linear Diophantine equation.
Greatest Common Divisor (gcd):
The gcd of two integers a and b, denoted as gcd(a, b), is the largest positive integer that divides both a and b without leaving a remainder.
Bezout's Identity:
According to Bezout's identity, for any two integers a and b, there exist integers x and y such that ax + by = gcd(a, b).
Relation to Linear Diophantine Equations:
Now, let's consider the linear Diophantine equation ax + by = c.
If a and b are relatively prime, i.e., gcd(a, b) = 1, then the equation becomes ax + by = 1. In this case, a solution exists for any integer values of x and y.
If a and b are not relatively prime, i.e., gcd(a, b) = d (where d > 1), then we can divide both sides of the equation by d to get (a/d)x + (b/d)y = c/d. Now, the equation is in the form of ax' + by' = c', where x' = x/d, y' = y/d, and c' = c/d. Since c' is a multiple of d, a solution exists if and only if c is a multiple of d.
Conclusion:
From the above explanation, we can conclude that option 'A' is true. A solution exists for the linear Diophantine equation ax + by = c if c is a multiple of gcd(a, b). This is because the gcd represents the common divisor that both a and b share, and any multiple of the gcd can be expressed as a linear combination of a and b.
A linear Diophantine equation is an equation of the form ax + by = c, where a, b, and c are integers, and x and y are variables. The goal is to find integer solutions for x and y that satisfy the equation.
Existence of Solutions:
In the given linear Diophantine equation ax + by = c, a solution exists if and only if c is a multiple of the greatest common divisor (gcd) of a and b.
Explanation:
To understand why option 'A' is true, let's consider the concept of the gcd and its relation to the solution of a linear Diophantine equation.
Greatest Common Divisor (gcd):
The gcd of two integers a and b, denoted as gcd(a, b), is the largest positive integer that divides both a and b without leaving a remainder.
Bezout's Identity:
According to Bezout's identity, for any two integers a and b, there exist integers x and y such that ax + by = gcd(a, b).
Relation to Linear Diophantine Equations:
Now, let's consider the linear Diophantine equation ax + by = c.
If a and b are relatively prime, i.e., gcd(a, b) = 1, then the equation becomes ax + by = 1. In this case, a solution exists for any integer values of x and y.
If a and b are not relatively prime, i.e., gcd(a, b) = d (where d > 1), then we can divide both sides of the equation by d to get (a/d)x + (b/d)y = c/d. Now, the equation is in the form of ax' + by' = c', where x' = x/d, y' = y/d, and c' = c/d. Since c' is a multiple of d, a solution exists if and only if c is a multiple of d.
Conclusion:
From the above explanation, we can conclude that option 'A' is true. A solution exists for the linear Diophantine equation ax + by = c if c is a multiple of gcd(a, b). This is because the gcd represents the common divisor that both a and b share, and any multiple of the gcd can be expressed as a linear combination of a and b.
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In linear diophantine equations of the form ax + by = c, where a, b, and c are integers, which of the following is true?a)A solution exists if c is a multiple of gcd(a, b).b)A solution exists if c is a prime number.c)A solution exists if a and b are prime numbers.d)A solution exists if c is an odd number.Correct answer is option 'A'. Can you explain this answer?
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In linear diophantine equations of the form ax + by = c, where a, b, and c are integers, which of the following is true?a)A solution exists if c is a multiple of gcd(a, b).b)A solution exists if c is a prime number.c)A solution exists if a and b are prime numbers.d)A solution exists if c is an odd number.Correct answer is option 'A'. Can you explain this answer? for Software Development 2025 is part of Software Development preparation. The Question and answers have been prepared according to the Software Development exam syllabus. Information about In linear diophantine equations of the form ax + by = c, where a, b, and c are integers, which of the following is true?a)A solution exists if c is a multiple of gcd(a, b).b)A solution exists if c is a prime number.c)A solution exists if a and b are prime numbers.d)A solution exists if c is an odd number.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Software Development 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In linear diophantine equations of the form ax + by = c, where a, b, and c are integers, which of the following is true?a)A solution exists if c is a multiple of gcd(a, b).b)A solution exists if c is a prime number.c)A solution exists if a and b are prime numbers.d)A solution exists if c is an odd number.Correct answer is option 'A'. Can you explain this answer?.
In linear diophantine equations of the form ax + by = c, where a, b, and c are integers, which of the following is true?a)A solution exists if c is a multiple of gcd(a, b).b)A solution exists if c is a prime number.c)A solution exists if a and b are prime numbers.d)A solution exists if c is an odd number.Correct answer is option 'A'. Can you explain this answer? for Software Development 2025 is part of Software Development preparation. The Question and answers have been prepared according to the Software Development exam syllabus. Information about In linear diophantine equations of the form ax + by = c, where a, b, and c are integers, which of the following is true?a)A solution exists if c is a multiple of gcd(a, b).b)A solution exists if c is a prime number.c)A solution exists if a and b are prime numbers.d)A solution exists if c is an odd number.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Software Development 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In linear diophantine equations of the form ax + by = c, where a, b, and c are integers, which of the following is true?a)A solution exists if c is a multiple of gcd(a, b).b)A solution exists if c is a prime number.c)A solution exists if a and b are prime numbers.d)A solution exists if c is an odd number.Correct answer is option 'A'. Can you explain this answer?.
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