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# help number theory

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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?

Jul 9, 2021

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The sum of the three smallest possible values of x is 858.

Jul 9, 2021
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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?

$$\begin{array}{|rcll|} \hline x&\equiv& 3 \pmod{4} \\ x&\equiv& 3 \pmod{5} \\ x&\equiv& 3 \pmod{6} \\ x&\equiv& 3 \pmod{7} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \text{sum} &=& \Big( 3+0\times \text{lcm}(4,~5,~6,~7) \Big) \\ && +\Big( 3+1\times \text{lcm}(4,~5,~6,~7) \Big) \\ && +\Big( 3+2\times \text{lcm}(4,~5,~6,~7) \Big) \\\\ \text{sum} &=& 3 + ( 3+420 )+( 3+2\times420 ) \\\\ \text{sum} &=& 3 + 423+843 \\ \text{sum} &=& 1269 \\ \hline \end{array}$$

Jul 10, 2021
edited by heureka  Jul 10, 2021