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When x is divided by each of 4, 5, and 6, remainders of 3, 4, and 5 (respectively) are obtained. What is the smallest possible positive integer value of x?

Aug 8, 2018

#1
+22358
+1

When x is divided by each of 4, 5, and 6, remainders of 3, 4, and 5 (respectively) are obtained.

What is the smallest possible positive integer value of x?

$$\begin{array}{|lrclcrcl|} \hline & x &\equiv& 3 \pmod 4 &\text{or} & x &\equiv& -1 \pmod 4 \\ & x &\equiv& 4 \pmod 5 &\text{or} & x &\equiv& -1 \pmod 5 \\ & x &\equiv& 5 \pmod 6 &\text{or} & x &\equiv& -1 \pmod 6 \\\\ \Rightarrow & x &\equiv& -1 \pmod{\text{lcm}(4,5,6)} \\ & x &\equiv& -1 \pmod{60} \\ & x &\equiv& -1+60 \pmod{60} \\ & \mathbf{x} & \mathbf{\equiv} & \mathbf{59 \pmod{60}} \\ \hline \end{array}$$

The smallest possible positive integer value x is 59

Aug 8, 2018
edited by heureka  Aug 8, 2018
edited by heureka  Aug 8, 2018

#1
+22358
+1

When x is divided by each of 4, 5, and 6, remainders of 3, 4, and 5 (respectively) are obtained.

What is the smallest possible positive integer value of x?

$$\begin{array}{|lrclcrcl|} \hline & x &\equiv& 3 \pmod 4 &\text{or} & x &\equiv& -1 \pmod 4 \\ & x &\equiv& 4 \pmod 5 &\text{or} & x &\equiv& -1 \pmod 5 \\ & x &\equiv& 5 \pmod 6 &\text{or} & x &\equiv& -1 \pmod 6 \\\\ \Rightarrow & x &\equiv& -1 \pmod{\text{lcm}(4,5,6)} \\ & x &\equiv& -1 \pmod{60} \\ & x &\equiv& -1+60 \pmod{60} \\ & \mathbf{x} & \mathbf{\equiv} & \mathbf{59 \pmod{60}} \\ \hline \end{array}$$

The smallest possible positive integer value x is 59

heureka Aug 8, 2018
edited by heureka  Aug 8, 2018
edited by heureka  Aug 8, 2018