What is the largest negative integer \(x\) satisfying \(24x \equiv 15 \pmod{1199}~?\)
+118608
What is the largest negative x integer satisfying \( 24x \equiv 15 \pmod{1199}~? \)
15 (mod1199) is equivalent to 15-1199- 1199k = -1184 - 1199k (mod1199)
24x = -1184 - 1199k
1199k+24x = -1184
there does not appear to be any common factors
Look for a solution to 1199k+24x=-1
1199=50*24-1
so
1199(1)+24(-50)=-1
and
1199(1184 )+24(-50*1184 )=-1184
the general solution is
1199(1184 -24a )+24(-50*1184 + 1199a )=-1184 where a is an integer
so I think I want
\(-50*1184+1199a<0\\ 1199a<58200\\ a<49.37\\ \text{but a is an integer so the biggest a is 49}\)
\(1199(1184-24*49) +24(-50*1184+1199*49)=-1184\\ 1184(8)+24(-449)=-1184\)
\( 24*-449 \equiv 15 \pmod{1199} \)
So I think that the negative integer with the smallest absolute value (that is the biggest one) is
\(x= -449\)
+26367 What is the largest negative integer \(x\) satisfying \(24x \equiv 15 \pmod{1199}~?\)
\(\begin{array}{|rclll|} \hline 24x &\equiv& 15 \pmod{1199} \\ \text{or} \\ 24x &=& 15 + 1199k \quad & | \quad k \in Z \\\\ \mathbf{x} &\mathbf{=}& \mathbf{\dfrac{15 + 1199k}{24}} \\\\ x &=& \dfrac{49\cdot 24k + 23k + 15 }{24} \\\\ x &=& 49k + \underbrace{ \dfrac{23k + 15 }{24} }_{=a} \\\\ && & a = \dfrac{23k + 15 }{24} \\\\ && & 24a = 23k + 15 \\\\ && & 23k = 24a - 15 \\\\ && & \mathbf{ k = \dfrac{24a - 15}{23} }\\\\ && & k = \dfrac{23a+a - 15}{23} \\\\ && & k = a + \underbrace{ \dfrac{a - 15}{23} }_{=b} \\\\ && && b = \dfrac{a - 15}{23} \\\\ && && 23b = a - 15 \\\\ && & & \mathbf{ a = 23b + 15} \\\\ && & \mathbf{ k = \dfrac{24(23b + 15) - 15}{23} }\\\\ && & k = \dfrac{24\cdot 23b + 24\cdot 15 - 15}{23} \\\\ && & k = \dfrac{24\cdot 23b + 23\cdot 15}{23} \\\\ && & \mathbf{k = 24b+15} \\\\ \mathbf{x} &\mathbf{=}& \mathbf{\dfrac{15 + 1199(24b+15)}{24}} \\\\ x & = & \dfrac{15 + 1199\cdot 24b+ 1199\cdot 15}{24} \\\\ x & = & \dfrac{1199\cdot 24b+ 1200\cdot 15}{24} \\\\ x & = & \dfrac{1199\cdot 24b+ 50\cdot 24 \cdot 15}{24} \\\\ x & = & 50\cdot 15 + 1199 b \\\\ x & = & 750 + 1199 b \quad & | \quad b = -1 \\\\ x & = & 750 - 1199 \\\\ \mathbf{ x } &\mathbf{ = } & \mathbf{-449} \\ \hline \end{array}\)
