Casting Out Nines/Examples/61,934 and 38,247

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Examples of Use of Casting Out Nines

We have:

\(\ds \dr {61 \, 934}\) \(=\) \(\ds 5\)
\(\ds \dr {38 \, 247}\) \(=\) \(\ds 6\)
\(\ds \leadsto \ \ \) \(\ds \dr {61 \, 934 + 38 \, 247}\) \(=\) \(\ds 2\)
\(\ds \dr {61 \, 934 - 38 \, 247}\) \(=\) \(\ds 8\)
\(\ds \dr {61 \, 934 \times 38 \, 247}\) \(=\) \(\ds 3\)


Proof

\(\ds 61 \, 934 + 38 \, 247\) \(=\) \(\ds 100 \, 181\)
\(\ds \leadsto \ \ \) \(\ds \dr {61 \, 934 + 38 \, 247}\) \(=\) \(\ds \dr {100 \, 181}\)
\(\ds \) \(=\) \(\ds \dr {1 + 0 + 0 + 1 + 8 + 1}\)
\(\ds \) \(=\) \(\ds \dr {11}\)
\(\ds \) \(=\) \(\ds 2\)
\(\ds \dr {61 \, 934 + 38 \, 247}\) \(=\) \(\ds \dr {5 + 6}\)
\(\ds \) \(=\) \(\ds \dr {11}\)
\(\ds \) \(=\) \(\ds 2\)


\(\ds 61 \, 934 - 38 \, 247\) \(=\) \(\ds 23 \, 687\)
\(\ds \leadsto \ \ \) \(\ds \dr {61 \, 934 - 38 \, 247}\) \(=\) \(\ds \dr {23 \, 687}\)
\(\ds \) \(=\) \(\ds \dr {2 + 3 + 6 + 8 + 7}\)
\(\ds \) \(=\) \(\ds \dr {26}\)
\(\ds \) \(=\) \(\ds 8\)
\(\ds \dr {61 \, 934 + 38 \, 247}\) \(=\) \(\ds \dr {5 - 6}\)
\(\ds \) \(=\) \(\ds \dr {-1}\)
\(\ds \) \(=\) \(\ds \dr 8\)
\(\ds \) \(=\) \(\ds 8\)


\(\ds 61 \, 934 \times 38 \, 247\) \(=\) \(\ds 2 \, 368 \, 789 \, 698\)
\(\ds \leadsto \ \ \) \(\ds \dr {61 \, 934 \times 38 \, 247}\) \(=\) \(\ds \dr {2 \, 368 \, 789 \, 698}\)
\(\ds \) \(=\) \(\ds \dr {2 + 3 + 6 + 8 + 7 + 8 + 9 + 6 + 9 + 8}\)
\(\ds \) \(=\) \(\ds \dr {2 + 3 + 6 + 8 + 7 + 8 + 6 + 8}\) Casting Out Nines
\(\ds \) \(=\) \(\ds \dr {48}\)
\(\ds \) \(=\) \(\ds \dr {4 + 8}\)
\(\ds \) \(=\) \(\ds \dr {12}\)
\(\ds \) \(=\) \(\ds 3\)
\(\ds \dr {61 \, 934 \times 38 \, 247}\) \(=\) \(\ds \dr {5 \times 6}\)
\(\ds \) \(=\) \(\ds \dr {30}\)
\(\ds \) \(=\) \(\ds 3\)

$\blacksquare$


Sources

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