Reversed number

Let mrm_r be the reflection of mm. For example, 1234r=4321 1234_r = 4321 .

The positive integer k has the property,

mN,km    kmr\forall m\in\mathbb{N},k|m \implies k|m_r.

Show that, k99 k \mid 99 .

Note by Kalpa Roy
2 years, 1 month ago

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1 vote

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Lets take 4 digit abcdabcd.

So k1000d+100c+10b+ak|1000d+100c+10b+a & k1000a+100b+10c+dk|1000a+100b+10c+d

    k999d+90c90b999a\implies k|999d+90c-90b-999a

    k9\implies k|9.

Again

k1000d+100c+10b+a+1000a+100b+10c+dk|1000d+100c+10b+a+1000a+100b+10c+d

    k1001d+110c+110b+1001a\implies k|1001d+110c+110b+1001a

    k11\implies k|11

Combining

k99k|99.

YOU CAN PROVE IT USING INDUCTION.

Md Zuhair - 2 years ago

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Not quite. Be careful of your implication signs. For example, if k999 k \mid 999 , must we have k1 k \mid 1 ?

Calvin Lin Staff - 2 years ago

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

Md Zuhair - 2 years ago

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can anyone please help me out with this problem

Kalpa Roy - 2 years, 1 month ago

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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

Md Zuhair - 2 years ago

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not correct

Michael Fitzgerald - 2 years ago

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

Md Zuhair - 2 years ago

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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.

Sathvik Acharya - 2 years ago

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