- #1

Math100

- 771

- 219

- Homework Statement
- Find the smallest integer ## a>2 ## such that ## 2\mid a, 3\mid (a+1), 4\mid (a+2), 5\mid (a+3), 6\mid (a+4) ##.

- Relevant Equations
- None.

Let ## a>2 ## be the smallest integer.

Then

\begin{align*}

&2\mid a\implies a\equiv 0\pmod {2}\implies a\equiv 2\pmod {2}\\

&3\mid (a+1)\implies a+1\equiv 0\pmod {3}\implies a\equiv -1\pmod {3}\implies a\equiv 2\pmod {3}\\

&4\mid (a+2)\implies a+2\equiv 0\pmod {4}\implies a\equiv -2\pmod {4}\implies a\equiv 2\pmod {4}\\

&5\mid (a+3)\implies a+3\equiv 0\pmod {5}\implies a\equiv -3\pmod {5}\implies a\equiv 2\pmod {5}\\

&6\mid (a+4)\implies a+4\equiv 0\pmod {6}\implies a\equiv -4\pmod {6}\implies a\equiv 2\pmod {6}.\\

\end{align*}

Observe that ## lcm(2, 3, 4, 5, 6)=60 ##.

Thus ## a\equiv 2\pmod {60}\implies a=62 ##.

Therefore, the smallest integer ## a>2 ## such that ## 2\mid a, 3\mid (a+1), 4\mid (a+2), 5\mid (a+3), 6\mid (a+4) ## is ## 62 ##.

Then

\begin{align*}

&2\mid a\implies a\equiv 0\pmod {2}\implies a\equiv 2\pmod {2}\\

&3\mid (a+1)\implies a+1\equiv 0\pmod {3}\implies a\equiv -1\pmod {3}\implies a\equiv 2\pmod {3}\\

&4\mid (a+2)\implies a+2\equiv 0\pmod {4}\implies a\equiv -2\pmod {4}\implies a\equiv 2\pmod {4}\\

&5\mid (a+3)\implies a+3\equiv 0\pmod {5}\implies a\equiv -3\pmod {5}\implies a\equiv 2\pmod {5}\\

&6\mid (a+4)\implies a+4\equiv 0\pmod {6}\implies a\equiv -4\pmod {6}\implies a\equiv 2\pmod {6}.\\

\end{align*}

Observe that ## lcm(2, 3, 4, 5, 6)=60 ##.

Thus ## a\equiv 2\pmod {60}\implies a=62 ##.

Therefore, the smallest integer ## a>2 ## such that ## 2\mid a, 3\mid (a+1), 4\mid (a+2), 5\mid (a+3), 6\mid (a+4) ## is ## 62 ##.