Smart Numbers

For a required smart number $n$ we have $n=17x$ for some natural number $x$ .

Also $n=17x \equiv 17 \pmod{100}$

$\Rightarrow$ $x \equiv 1 \pmod{100}$

$\Rightarrow$ $x=1, 101, 201, 301......$

Thus the required sum is $17*(1+101+201+301+401)=17085$ .

Any smart number will be of form $100x+17$ .

Since $17|\left( 100x+17 \right)$ and $17|17$ , we can say that $17|100x$ .

Since $\gcd\left( 100,17 \right) =1$ , we can also say that $17|x$ . Now we can solve the problem using this knowledge.

As we are looking for smallest smart numbers, value of $x$ will be 0, 17, 34, 51 and 68. Hence the numbers are

$100\left( 0 \right) +17=17\\ 100\left( 17 \right) +17=1717\\ 100\left( 34 \right) +17=3417\\ 100\left( 51 \right) +17=5117\\ 100\left( 68 \right) +17=6817$

Sum = 17085

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Note: $a|b$ means $a$ divides $b$ . Eg. $17|34$ and $3|6$ .