+1833 $$Several congruences are listed below and labeled with powers of 2. Some of the congruences are true and some are false. Find the sum of the labels for the congruences that are true.
(1) $118 \equiv 25 \pmod{13}$
(2) $2401 \equiv 147 \pmod{49}$
(4) $183 \equiv 291 \pmod{6}$
(8) $2701 \equiv 14393 \pmod{8}$
(16) $493 \equiv 873 \pmod{10}$
(32) $4113 \equiv 396 \pmod{9}$$$Several congruences are listed below and labeled with powers of 2. Some of the congruences are true and some are false. Find the sum of the labels for the congruences that are true. (1) $118 \equiv 25 \pmod{13}$ (2) $2401 \equiv 147 \pmod{49}$ (4) $183 \equiv 291 \pmod{6}$ (8) $2701 \equiv 14393 \pmod{8}$ (16) $493 \equiv 873 \pmod{10}$ (32) $4113 \equiv 396 \pmod{9}$
+26367 $$\\ \small{\text{
Several congruences are listed below and labeled with powers of 2.
}}\\
\small{\text{
Some of the congruences are true and some are false.
}}\\
\small{\text{
Find the sum of the labels for the congruences that are true.
}}\\$$
$$\small{\text{$
\begin{array}{rrcl|rcl|rcl|r}
(1) & 118 &\equiv & 25 \pmod{13} & 118 \pmod{13} &\equiv& 1 & 25\pmod{13} &\equiv& 12 & \mathrm{false}\\
(2) & 2401 &\equiv & 147 \pmod{49} & 2401 \pmod{49} &\equiv& 0 & 147\pmod{49} &\equiv& 0 & \mathrm{true}\\
(4) & 183 &\equiv & 291 \pmod{6} & 183 \pmod{6} &\equiv& 3 & 291\pmod{6} &\equiv& 3 & \mathrm{true}\\
(8) & 2701 &\equiv & 14393 \pmod{8} & 2701 \pmod{8} &\equiv& 5 & 14393\pmod{8} &\equiv& 1 & \mathrm{false}\\
(16)& 493 &\equiv & 873 \pmod{10} & 493 \pmod{10} &\equiv& 3 & 873\pmod{10} &\equiv& 3 & \mathrm{true}\\
(32)& 4113 &\equiv & 396 \pmod{9} & 4113 \pmod{9} &\equiv& 0 & 396\pmod{9} &\equiv& 0 & \mathrm{true}\\
\end{array}
$}}$$
sum of the labels for the conguences that are true: (2)+(4)+(16)+(32) = 54
+26367 $$\\ \small{\text{
Several congruences are listed below and labeled with powers of 2.
}}\\
\small{\text{
Some of the congruences are true and some are false.
}}\\
\small{\text{
Find the sum of the labels for the congruences that are true.
}}\\$$
$$\small{\text{$
\begin{array}{rrcl|rcl|rcl|r}
(1) & 118 &\equiv & 25 \pmod{13} & 118 \pmod{13} &\equiv& 1 & 25\pmod{13} &\equiv& 12 & \mathrm{false}\\
(2) & 2401 &\equiv & 147 \pmod{49} & 2401 \pmod{49} &\equiv& 0 & 147\pmod{49} &\equiv& 0 & \mathrm{true}\\
(4) & 183 &\equiv & 291 \pmod{6} & 183 \pmod{6} &\equiv& 3 & 291\pmod{6} &\equiv& 3 & \mathrm{true}\\
(8) & 2701 &\equiv & 14393 \pmod{8} & 2701 \pmod{8} &\equiv& 5 & 14393\pmod{8} &\equiv& 1 & \mathrm{false}\\
(16)& 493 &\equiv & 873 \pmod{10} & 493 \pmod{10} &\equiv& 3 & 873\pmod{10} &\equiv& 3 & \mathrm{true}\\
(32)& 4113 &\equiv & 396 \pmod{9} & 4113 \pmod{9} &\equiv& 0 & 396\pmod{9} &\equiv& 0 & \mathrm{true}\\
\end{array}
$}}$$
sum of the labels for the conguences that are true: (2)+(4)+(16)+(32) = 54
