Exercise · ZW-LINEAR-DIOPHANTINE-EQUATION-002 · ★☆☆ Basic
Which statement correctly describes when \(ax+by=c\) has integer solutions?
Choose one option.
Show answer
- B. When \(\gcd(a,b)\) divides \(c\)
Explanation
The exact condition is that the gcd of the coefficients divides the constant term. Requiring only that \(a\) divides \(c\) or that \(c\) is even are insufficient conditions.
Review the concept: Linear Diophantine equation
A linear Diophantine equation is \(ax+by=c\) with integer coefficients, where we seek integer pairs \(x,y\) as solutions.
\[x=x_0+\frac{b}{d}t,\quad y=y_0-\frac{a}{d}t,\quad t\in\mathbb{Z}\]
More practice problems: linear diophantine equation
- ZW-LINEAR-DIOPHANTINE-EQUATION-001 What is the greatest common divisor of 12 and 18, which determines whether 12x+18y=c has a solution?
- ZW-LINEAR-DIOPHANTINE-EQUATION-003 What is the name of the theorem guaranteeing that d=gcd(a,b) can be written as ax+by with integers…
- ZW-LINEAR-DIOPHANTINE-EQUATION-004 What is the greatest common divisor of 15 and 35?
- ZW-LINEAR-DIOPHANTINE-EQUATION-005 Does the equation 4x+6y=5 have integer solutions?
- ZW-LINEAR-DIOPHANTINE-EQUATION-006 If 7(3)+5(-4)=1, what is a particular solution of 7x+5y=1?
- ZW-LINEAR-DIOPHANTINE-EQUATION-007 Given the particular solution (x_0,y_0)=(2,-1) of 4x+6y=2, what is the general solution with t∈ℤ?
- ZW-LINEAR-DIOPHANTINE-EQUATION-008 Let 9x+15y=12. What is the particular solution obtained from 3=9(2)+15(-1)?
- ZW-LINEAR-DIOPHANTINE-EQUATION-009 For which value of c does the equation 10x+4y=c have an integer solution?
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