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Show that the five digit number $abcde$ is congruent (mod $11$) to $(a + c + e) - (b + d)$

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closed as off-topic by Martin R, N. F. Taussig, Javi, José Carlos Santos, Leucippus Apr 22 at 15:04

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  • $\begingroup$ Observe that $$abcde = 10000 \times a + 1000 \times b + 100 \times c + 10 \times d + e.$$ $\endgroup$ – Dbchatto67 Apr 22 at 7:47
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Hint: Write your number in the form $$e+10d+10^2c+10^3b+a10^4$$ and note that $$10\equiv -1\mod 11$$ $$10^2\equiv 1 \mod 11$$ $$10^3\equiv -1\mod 11$$ $$10^4\equiv 1 \mod 11$$

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Observe that $$abcde = 10000 \times a + 1000 \times b + 100 \times c + 10 \times d + e.$$

So $$\begin{align*} abcde - (a+c+e) + (b+d) & \equiv 9999 \times a + 1001 \times b + 99 \times c + 11 \times d\ (\text {mod}\ 11) \\ & \equiv 0\ (\text {mod}\ 11) \end{align*}$$

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