Find the missing digits in the calculations below

[Math100]

Homework Statement

Working modulo $9$ or $11$, find the missing digits in the calculations below:
$51840\cdot 273581=1418243\chi 040$.

Relevant Equations

None.

First, consider modulo $9$.
Then $9\mid (5+1+8+4+0)\implies 9\mid 18\implies 9\mid 51840$.
Since $51840\cdot 273581=1418243\chi 040$, it follows that $9\mid 1418243\chi 040$.
This means $9\mid (1+4+1+8+2+4+3+\chi+0+4+0)\implies 9\mid (27+\chi)\implies 9\mid\chi$.
Thus $\chi=0$ or $\chi=9$.
Next, consider modulo $11$.
Then $11\mid (1-8+5-3+7-2)\implies 11\mid 0\implies 11\mid 273581$.
Since $51840\cdot 273581=1418243\chi 040$, it follows that $11\mid 1418243\chi 040$.
This means $11\mid (0-4+0-\chi+3-4+2-8+1-4+1)\implies 11\mid (-13-\chi)\implies 11\mid (13+\chi)$.
Thus $\chi=9$.
Therefore, $\chi=9$.

 

[SammyS]

Homework Statement:: Working modulo $9$ or $11$, find the missing digits in the calculations below:
$51840\cdot 273581=1418243\chi 040$.
Relevant Equations:: None.

First, consider modulo $9$.
Then $9\mid (5+1+8+4+0)\implies 9\mid 18\implies 9\mid 51840$.
Since $51840\cdot 273581=1418243\chi 040$, it follows that $9\mid 1418243\chi 040$.
This means $9\mid (1+4+1+8+2+4+3+\chi+0+4+0)\implies 9\mid (27+\chi)\implies 9\mid\chi$.
Thus $\chi=0$ or $\chi=9$.
Next, consider modulo $11$.
Then $11\mid (1-8+5-3+7-2)\implies 11\mid 0\implies 11\mid 273581$.
Since $51840\cdot 273581=1418243\chi 040$, it follows that $11\mid 1418243\chi 040$.
This means $11\mid (0-4+0-\chi+3-4+2-8+1-4+1)\implies 11\mid (-13-\chi)\implies 11\mid (13+\chi)$.
Thus $\chi=9$.
Therefore, $\chi=9$.

Looks good.

 

[Delta2]

Hmm so it seems to me you use two lemmas that would be good if you had explicitly mentioned them:

1. 9 divides a number if and only if it divides the sum of its digits in base 10
2. 11 divides a number if and only if it divides the alternating sum and difference of its digits in base 10