Prove that the last two digits are 28

Prove that if nn is an odd positive integer, then the last two digits of 22n(22n+11)2^{2n}(2^{2n+1}-1) in base 1010 are 2828.

Note by Finn Hulse
5 years, 3 months ago

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Hint:Induction

Bogdan Simeonov - 5 years, 3 months ago

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Okay... Can you explain?

Finn Hulse - 5 years, 3 months ago

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When n=1, the result is 28.Suppose for some n the condition is true.

Multiply by 16^2 and add 15.22n+415.2^{2n+4} and you will get the next such number.So the new number will be congruent to

256.28+15.22n+468+15.22n+4(mod100)256.28+15.2^{2n+4}\equiv 68+15.2^{2n+4}(mod100)

But 15.22n15.2^{2n} for odd numbers n will be congruent to 60 modulo 100 (that's not hard to see), so we have proven in inductively.

Bogdan Simeonov - 5 years, 3 months ago

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@Bogdan Simeonov Oh, I guess so! Great job Bogdan. :D

Finn Hulse - 5 years, 3 months ago

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@Finn Hulse You probably have no idea how to pronounce my name :D

Bogdan Simeonov - 5 years, 3 months ago

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@Bogdan Simeonov Probably. How? :D

Finn Hulse - 5 years, 3 months ago

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@Finn Hulse Try using the speech feature on it.It sounds stupid :D

Bogdan Simeonov - 5 years, 3 months ago

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@Bogdan Simeonov HAHA yeah. But how is your name actually pronounced?

Finn Hulse - 5 years, 3 months ago

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@Finn Hulse I believe it should be pronounced like this: pronounce log, but with a B instead of an l, then say the name Dan :D

Bogdan Simeonov - 5 years, 3 months ago

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@Bogdan Simeonov Oh, that's what I thought. :D

Finn Hulse - 5 years, 3 months ago

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