How many triples (a,b,c) of even positive integers satisfy \(a^3+b^2+c\le 50\)?
+981 Note: this is solution is incorrect, since I missed the less than, I thought it was just an equal sign.
We first need to determine the range of for \(a\):
\(1^3=1,\\ 2^3=8,\\ 3^3=27,\\ 4^3=64.\\ \Rightarrow{a=2} \)
\(a=2, \text{since a is even}\\ \Rightarrow2^3+b^2+c=50\\ b^2+c=42\)
Repeating our process, b can be 2, 4, or 6
There is a fixed value for c for each value of b.
Therefore, there are 3 triples that satisfy \(a^3+b^2+c=50\)
+26367 How many triples (a,b,c) of even positive integers satisfy
\(a^3+b^2+c\le 50\)
a^3+b^2+c\le 50 ?
\(\begin{array}{|r|r|r|r|r|} \hline & a & b & c & a^3+b^2+c \le 50 \\ \hline 1 & 2&2&2 & 14 \\ 2 & 2&2&4 & 16 \\ 3 & 2&2&6 & 18 \\ 4 & 2&2&8 & 20 \\ 5 & 2&2&10 & 22 \\ 6 & 2&2&12 & 24 \\ 7 & 2&2&14 & 26 \\ 8 & 2&2&16 & 28 \\ 9 & 2&2&18 & 30 \\ 10 & 2&2&20 & 32 \\ 11 & 2&2&22 & 34 \\ 12 & 2&2&24 & 36 \\ 13 & 2&2&26 & 38 \\ 14 & 2&2&28 & 40 \\ 15 & 2&2&30 & 42 \\ 16 & 2&2&32 & 44 \\ 17 & 2&2&34 & 46 \\ 18 & 2&2&36 & 48 \\ 19 & 2&2&38 & 50 \\ \hline 20 & 2& 4& 2 & 26 \\ 21 & 2& 4& 4 & 28 \\ 22 & 2& 4& 6 & 30 \\ 23 & 2& 4& 8 & 32 \\ 24 & 2& 4& 10 & 34 \\ 25 & 2& 4& 12 & 36 \\ 26 & 2& 4& 14 & 38 \\ 27 & 2& 4& 16 & 40 \\ 28 & 2& 4& 18 & 42 \\ 29 & 2& 4& 20 & 44 \\ 30 & 2& 4& 22 & 46 \\ 31 & 2& 4& 24 & 48 \\ 32 & 2& 4& 26 & 50 \\ \hline 33 & 2& 6& 2 & 46 \\ 34 & 2& 6& 4 & 48 \\ 35 & 2& 6& 6 & 50 \\ \hline \end{array}\)
\(\text{ $\mathbf{35}$ triples $(a,b,c)$ of even positive integers satisfy $a^3+b^2+c \le 50$ }\)
