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Let $m_r$ be the reflection of $m$. For example, $1234_r = 4321$.

The positive integer k has the property,

$\forall m\in\mathbb{N},k|m \implies k|m_r$.

Show that, $k \mid 99$.

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Lets take 4 digit $abcd$.

So $k|1000d+100c+10b+a$ & $k|1000a+100b+10c+d$

$\implies k|999d+90c-90b-999a$

$\implies k|9$.

Again

$k|1000d+100c+10b+a+1000a+100b+10c+d$

$\implies k|1001d+110c+110b+1001a$

$\implies k|11$

Combining

$k|99$.

YOU CAN PROVE IT USING INDUCTION.

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Not quite. Be careful of your implication signs. For example, if $k \mid 999$, must we have $k \mid 1$?

Oops no.. yes.. thats true... ok.. thanks sir

can anyone please help me out with this problem

What have you tried? Where are you stuck?

Its easy I guess.. will post a soln. I didnt did it with pen and paper. Idk if I am correct becoz i did it in my mind

not correct

@Michael Fitzgerald – Ya... Calvin sir told it before also... I know its wrong

Hint: Try to use the fact that even when the number is reflected it's sum of the digits still remain the same.

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$</code> ... <code>$</code>...<code>."> Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in $</span> ... <span>$ or $</span> ... <span>$ to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestLets take 4 digit $abcd$.

So $k|1000d+100c+10b+a$ & $k|1000a+100b+10c+d$

$\implies k|999d+90c-90b-999a$

$\implies k|9$.

Again

$k|1000d+100c+10b+a+1000a+100b+10c+d$

$\implies k|1001d+110c+110b+1001a$

$\implies k|11$

Combining

$k|99$.

YOU CAN PROVE IT USING INDUCTION.

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Not quite. Be careful of your implication signs. For example, if $k \mid 999$, must we have $k \mid 1$?

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Oops no.. yes.. thats true... ok.. thanks sir

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can anyone please help me out with this problem

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What have you tried? Where are you stuck?

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Its easy I guess.. will post a soln. I didnt did it with pen and paper. Idk if I am correct becoz i did it in my mind

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not correct

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Hint: Try to use the fact that even when the number is reflected it's sum of the digits still remain the same.

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