Find an integer having the remainders ## 2, 3, 4, 5 ##.

[Math100]

Homework Statement

Find an integer having the remainders $2, 3, 4, 5$ when divided by $3, 4, 5, 6$, respectively. (Bhaskara, born $1114$ ).

Relevant Equations

None.

Let $x$ be an integer.
Then $x\equiv 2\pmod {3}, x\equiv 3\pmod {4}, x\equiv 4\pmod {5}$ and $x\equiv 5\pmod {6}$.
This means
\begin{align*}
&x\equiv 2\pmod {3}\implies x+1\equiv 3\pmod {3}\implies x+1\equiv 0\pmod {3},\\
&x\equiv 3\pmod {4}\implies x+1\equiv 4\pmod {4}\implies x+1\equiv 0\pmod {4},\\
&x\equiv 4\pmod {5}\implies x+1\equiv 5\pmod {5}\implies x+1\equiv 0\pmod {5},\\
&x\equiv 5\pmod {6}\implies x+1\equiv 6\pmod {6}\implies x+1\equiv 0\pmod {6}.\\
\end{align*}
Observe that $lcm(3, 4, 5, 6)=60$.
Thus $x+1\equiv 0\pmod {60}\implies x\equiv -1\pmod {60}\implies x\equiv 59\pmod {60}$.
Therefore, the integer is $59$.

 

[fresh_42]

Correct. It looks somehow familiar. Have we had $59$ as a solution before?

 

[Math100]

Correct. It looks somehow familiar. Have we had $59$ as a solution before?

I don't remember. It does not look familiar to me, though. Interesting to know that.