How do I solve the linear diophantine equation [math]3x + 6y +5z = 7[/math]?
Consider the equation[math]\begin{align}\displaystyle 3x+6y+5z-7=0\end{align}\tag*{}[/math]In 3D space, this represents a plane with normal vector [math]\widehat{v}(3,6,5)[/math]. Bearing this in mind, here’s what we are going to do:Find a point on the plane with integer coordinatesFind two non-parallel vectors with integer components that lie on the planeExpress any other point on the plane with integer coordinates as a linear combination of these two vectors applied to our starting pointStep 1Trying some values for [math]x[/math], [math]y[/math] and [math]z[/math] we quickly find the point [math]P(-1,0,2)[/math] that lies in the plane and has integer coordinates.Step 2The vector [math]\hat{v}(3,6,5)[/math] is normal to the plane; therefore, any vector parallel to the plane should be perpendicular to [math]\hat{v}[/math]. So, if [math]\widehat{w}[/math] lies in the plane, then [math]\hat{v}\cdot{\hat{w}}=0[/math]. After some attempts (e.g for both vectors [math]\hat{w_a}[/math] and [math]\hat{w_b}[/math] that lie on the plane, set a component to [math]0[/math] and choose the other two components in such a way as to make the scalar products [math]\hat{w_a}\cdot{\hat{v}}[/math] and [math]\hat{w_b}\cdot{\hat{v}}[/math] equal to [math]0[/math]) we quickly come up with [math]\hat{w_a}(5,0,-3)[/math] and [math]\hat{w_b}(0,5,-6)[/math].Step 3Let [math]\hat{u}(-1,0,2)[/math] be the positional vector of the first point we found. Any other point with integer coordinates (i.e its positional vector) can be expressed as[math]\begin{align}\displaystyle \hat{u}+k\cdot{\hat{w_a}}+h\cdot{\hat{w_a}}\end{align}\tag*{}[/math]Hence the solution[math]\begin{align}\displaystyle\begin{cases}x=-1+5k\\y=5h\\z=2-3k-6h\end{cases}\end{align}\tag*{}[/math]with [math]k,h\in{\Z^2}[/math].
What would be the next term in the following series of 13, 17, 25, 32, and 37?
The guy nailed it...it all depends in what relationship you think about... which chapter is this question from?(Edit)The answer is 47.. the difference between the numbers is equal to the sum of the digits of the first number...For example:13 and 17 are the first two terms,1+3=4And17-13=4.. it is true for all the terms, so the last term is 47..How?3+7=10..10+37=47.that is the answer..
Pattern Recognition: Which one of the numbers does not belong in the following series: 1 - 2 - 5 - 10 - 13 - 26 - 29 - 48?
481 2 5 10 13 26 29 481 * 2 = 2 + 3 + 5 * 2 = 10 + 3 = 13 * 2 = 26 + 3 = 29 * 2 = 5848 has no place in the series according to the pattern that I found.