Japan | A Beautiful Olympiads Exponential Trick

Japan | A Beautiful Olympiads Exponential Trick | No Calculator Allowed

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16 күн бұрын

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Пікірлер: 8

@priyanshujha008 14 күн бұрын

AMAZING BRO KEEP IT UP

@adb012 13 күн бұрын

I think I can multiply 3 by itself 13 times in less time. Another way: 3^13-1=a ==> (3^13-3)/3=a/3 ==> 3^12-1=a/3, but 13^12 is 1000000000000 in base 3 (that's (10 base 3)^(12 base 10), a 1 with 12 zeros. So 3^12-1 = (1000000000000-1) base 3 = 222222222222 base 3 (twelve 2s, because the 2 is one less than the base, just like the 9 in base 10). Because that's a/3, to recover a we need to multiply by 3 which is by 10 base 3 so that's 2222222222220 base 3. That answer is already perfectly correct and I would be happy finishing the problem there, but if you prefer that's also 1594320 in base 10.

@timirbiswas3834 13 күн бұрын

Yes, u r right, excellently done.

@duckyoutube6318 12 күн бұрын

Interesting. I never thought of using other bases. Thank you for posting your technique. You have expanded my math toolkit.

@adb012 12 күн бұрын

@@duckyoutube6318 ... I never thought of that before either!!! Just learned it from a Stand-up Math video (Matt Parker's KZfaq channel) where he uses this to explain why b^n-1 is always divisible by b-1. b^n in base n will always be 10000 (n zeros) so when you subtract n it will be the equivalent to 99999 (n nines, or actually n times the bigger digit in the base-b system), and b-1 would be 9 (or actually the biggest digit in the base-b system, since since b will be always 10 in base b), and of course that 99999 will be divisible by 9!!! Go check it out, it's a great video!!!

@adb012 12 күн бұрын

@@duckyoutube6318 ... My only contribution to that was to expand it to the case where you have b^c-b^d, where d

@adb012 12 күн бұрын

Here, let me do it again with this insight: 3^13-3 = (10 base 3)^13 - (10 base 3) = 10000000000000 base 3 - 10 base 3 = 2222222222220 base 3. Done!