Question

Consider equation $I:x+y+z=46$ where $x,y$, and $z$ are positive integers, and equation II: $x+y+z+w=46$, where $x,y,z$ and $w$ are positive integers. Then

• A. I can be solved in consecutive integers
• B. I can be solved in consecutive even integers
• C. II can be solved in consecutive integers
• D. II can be solved in consecutive even integers
• E. II can be solved in consecutive odd integers

Answer 1

Answer: (C)

II can be solved in consecutive integers

For $\left(a\right)$ we have $n+n+1+n+2=3n+3-46$, an impossibility for integral $n$.
For $\left(b\right)$ we have $2n+2n+2+2n+4=6n+6=46$, an impossibility for integral $n$.
For $\left(c\right)$ we have $n+n+1+n+2+n+3=4n+6=46$, solvable for integral $n$.
For $\left(d\right)$ we have $2n+2n+2+2n+4+2n+6=8n+12=46$, an impossibility for integral $n$.
For $\left(e\right)$ we have $2n+1+2n+3+2n+5+2n+7=8n+16=46$, an impossibility for integral $n$.