When the positive integer x is divided by each of 4, 5, 6, and 7, it has a remainder of 3. What is the sum of the three smallest possible values of x?
+26367 When the positive integer x is divided by each of 4, 5, 6, and 7,
it has a remainder of 3.
What is the sum of the three smallest possible values of x?
\(\begin{array}{|rcll|} \hline x&\equiv& 3 \pmod{4} \\ x&\equiv& 3 \pmod{5} \\ x&\equiv& 3 \pmod{6} \\ x&\equiv& 3 \pmod{7} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \text{sum} &=& \Big( 3+0\times \text{lcm}(4,~5,~6,~7) \Big) \\ && +\Big( 3+1\times \text{lcm}(4,~5,~6,~7) \Big) \\ && +\Big( 3+2\times \text{lcm}(4,~5,~6,~7) \Big) \\\\ \text{sum} &=& 3 + ( 3+420 )+( 3+2\times420 ) \\\\ \text{sum} &=& 3 + 423+843 \\ \text{sum} &=& 1269 \\ \hline \end{array}\)
