alphabet soup

It is a known fact that the number 142857 gives a permutation of itself when multiplied by 1, 2, 3, 4, 5, 6, 8, 9. 142857 is the first 6 digit chunk after the decimal point in the decimal representation of 1/7. Cool number!

It is given that $N \times 5 = P$ as follows: :

$a\quad b\quad c\quad d\quad e\quad f$

$\times \quad \quad \quad \quad \quad \quad \quad 5$

$- - - - - - - - -$

$f\quad a\quad b\quad c\quad d\quad e$

From the above, it can be seen that $a=1$ , since $a\ne 0$ and if $a>1$ then $P$ will be $7$ digits.

For the $4^{th}$ digit of $P$ , $a$ , to be $1$ , $b$ must be even, so that $b\times 5 = \Box 0$ . If $b$ is odd, then $c$ needs to be either $12$ or $13$ which is unacceptable. Since $b$ is even, $c$ can only be $2$ or $3$ .

It is obvious that the last digit of $P$ , $e$ , can either be $0$ or $5$ . Assuming $e=0$ , then $f$ is even and $e\times 5$ has no carry-forward, therefore, $d\times 5 = \Box c$ . Since $e=0$ , $c$ must be $5$ and $d$ must be odd. Since $a=1$ , $d$ can only be $3$ and $f = 6$ (if $d\le 5$ then $f\le 10$ ). If $d=3$ and $c=5$ in $N$ , then $b$ in $P$ is $6$ which is unacceptable because $f = 6$ . Therefore, the assumption of $e=0$ is wrong. Then $e=5$ .

Now we have $a=1$ and $e=5$ . Since $e=5$ , $f$ is odd. Since $a=1$ , $f$ in $P$ must be larger then $5$ , that is $f=7$ or $9$ . Assuming $f=7$ , then:

$abcd57\times 5 = fabc85 \quad \Rightarrow d=8$

$abc857\times 5 = fab285 \quad \Rightarrow c=2$

$ab2857\times 5 = fa4285 \quad \Rightarrow b=4$

$a42857\times 5 = f14285 \quad \Rightarrow a=1$ , which is correct.

$142857\times 5 = 714285 \quad \Rightarrow a=1, b=4, c=2,d=8, e=5, f=7$

$\Rightarrow a+b+c+d+e+f=1+4+2+8+5+7=\boxed{27}$

$5 \overline{abcdef} = \overline{fabcde}$ $\Rightarrow 500000a + 50000b + 5000c + 500d + 50e = 100000f + 10000a + 1000b + 100c + 10d + e$

$\vdots$

$\Rightarrow 49 \overline{abcde} = 99995f$ $\Rightarrow \overline{abcde} = \frac{99995}{49}f = \frac{14285}{7}f$

Since $f$ and $\overline{abcde}$ are both integers , $f = 7$ , $\overline{abcde} = 14285$ , then you sum the digits up.