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I have to prove if the result of multiplication of 2 numbers is correct using the rule that a number is congruent to the sum of its numbers $\mod 9$. I am not really good in explaining so I will give an example:

I have to find out if $xy = z$ using $a \equiv b \pmod 9$ , where $b$ equals the sum of $a's$ digits. I would really appreciate some help, since I have no idea how to even start.

Thanks in advance

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    where are the numbers? โ€“ 2017-01-06
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    You may want to take a look at [the Wikipedia entry for casting out nines](https://en.wikipedia.org/wiki/Casting_out_nines). โ€“ 2017-01-06
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    The equation that I should verify for is the following : 54321 ยท 98765 = 5363013565 โ€“ 2017-01-07

1 Answers 1

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Hint:

For checking for $54321\cdot 98765 = 5363013565$ you should take the digital root of the product of digital roots of $x$ and $y$ respectively and compare it with the digital root of $z$.

This is:

$d(d(x)\cdot d(y)) = d(z)$

Thus:

$$54321 \equiv 6 \pmod 9$$

$$98765 \equiv 8 \pmod 9$$

$$5363013565 \equiv 1 \pmod 9$$

Since $6\cdot 8 \equiv 3 \pmod 9$ we can conclude that $xy \neq z$ for these values of $x,y,z$.