Exercise · ZW-LINEAR-DIOPHANTINE-EQUATION-001 · ★☆☆ Basic
What is the greatest common divisor of \(12\) and \(18\), which determines whether \(12x+18y=c\) has a solution?
Choose one option.
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- C. \(6\)
Explanation
The greatest common divisor of \(12\) and \(18\) is \(6\). Confusing it with \(3\) or with \(36\) are common errors when computing it.
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-002 Which statement correctly describes when ax+by=c has integer solutions?
- 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?