+885 1. What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 5?
2. What is the minimum value of the expression \(2x^2+3y^2+8x-24y+62\) for real \(x\) and \(y\)?
+26393 1. What is the smallest whole number that has
a remainder of 1 when divided by 4,
a remainder of 1 when divided by 3,
and a remainder of 2 when divided by 5?
\(\begin{array}{|rcll|} \hline x &\equiv& 1 \pmod{4} \\ x &\equiv& 1 \pmod{3} \\ x &\equiv& 2 \pmod{5} \\ \hline \end{array}\)
\(\small{ \begin{array}{|rcll|} \hline x &=& 1\cdot 3 \cdot 5 \cdot \frac{1}{3 \cdot 5}\pmod{4} + 1\cdot 4 \cdot 5 \cdot \frac{1}{4 \cdot 5}\pmod{3} + 2\cdot 4 \cdot 3 \cdot \frac{1}{4 \cdot 3}\pmod{5} + 4\cdot 3 \cdot 5 \cdot n \ |\ n\in Z \\ \hline \end{array} }\)
\(\begin{array}{|rcll|} \hline && \frac{1}{3 \cdot 5}\pmod{4} \\ &\equiv& (3 \cdot 5 )^{\varphi(4)-1} \pmod{4} \quad & | \quad \varphi(4) =4\cdot (1-\frac12) = 2 \\ &\equiv& 15^{1} \pmod{4} \\ &\equiv& 15 \pmod{4} \\ &\equiv& -1 \pmod{4} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline && \frac{1}{4 \cdot 5}\pmod{3} \\ &\equiv& (4 \cdot 5 )^{\varphi(3)-1} \pmod{3} \quad & | \quad \varphi(3) = 2 \\ &\equiv& 20^{1} \pmod{3} \\ &\equiv& 20 \pmod{3} \\ &\equiv& -1 \pmod{3} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline && \frac{1}{4 \cdot 3}\pmod{5} \\ &\equiv& (3 \cdot 5 )^{\varphi(4)-1} \pmod{4} \quad & | \quad \varphi(4) =4\cdot (1-\frac12) = 2 \\ &\equiv& 15^{3} \pmod{5} \quad & | \quad 12 \pmod{5} \equiv 2 \pmod{5} \\ &\equiv& 2^{3} \pmod{5} \\ &\equiv& 8 \pmod{5} \\ &\equiv& 3 \pmod{5} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline x &=& 15 \cdot (-1) + 20 \cdot (-1) + 24 \cdot 3 + 60 n \quad|\quad n\in Z \\ x &=& -15-20+72+60n \quad|\quad n\in Z \\ \mathbf{x} & \mathbf{=} & \mathbf{37+60n \quad|\quad n\in Z } \\ \hline \end{array}\)
\(\mathbf{x_{min} = 37 \quad | \quad n=0} \)
The smallest whole number is 37
+26393 2. What is the minimum value of the expression
2x^2+3y^2+8x-24y+62 for real \(x\) and \(y\)?
\(\begin{array}{|rcll|} \hline && 2x^2+3y^2+8x-24y+62 \\ &=& 2x^2+8x +3y^2-24y+62 \\ &=& 2(x^2+4x) +3(y^2-8y) + 62 \\ &=& 2[(x+2)^2-4] +3[(y-4)^2-16] + 62 \\ &=& 2(x+2)^2-8 +3(y-4)^2-48 + 62 \\ &=& 2(x+2)^2+3(y-4)^2-8 -48 + 62 \\ &=& 2(x {\color{red}+2})^2+3(y{\color{red}-4})^2 + 6\\ \hline \end{array}\)
We find global minimum:
\(\min\{ 2 x^2 + 3 y^2 + 8 x - 24 y + 62 \} = 6 \quad \text{at} \quad (x, y) = ( -2, 4)\)
