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Wrong puzzle !Should be : 1999^1999and also hint is A = B = C = D (mod 9)
btw the answer is 1
B=1+9+9+9+1+9+9+9=38C=3+8=11D=1+1=2
Hi all,the question is 1999^1999sorry for the mistake
Ok, that makes it a nicer altho' harder problem.The series 1999^1, 1999^2, 1999^3, 1999^4 all have a digital root of 1; eg 1999^4==1 (Mod 9). I'm guessing that 3 cycles of extracting the digital root of the thousands of digits of 1999^1999 is sufficient to reach the digital root; ie 1.
I like your writing style. Nice blog.
Let E be the sum of digits of D.Following the hint above (A, B, C, D and E are all equivalent mod 9) and using 1999 = 222*9 + 1 we get that E is equivalent to 1^1999 = 1 mod 9.Let n(x) denote the number of digits of x. We haven(A) <= 1999*log_{10}(1999) + 1 <= 1999*4 + 1 <= 8000.Therefore, B <= 9*n(A) <= 72000 which implies n(B) <= 5.On the same way we get:C <= 9*n(B) <= 45,n(C) <= 2, andD <= 9*n(C) <= 18.Every natural number in [1, 18] has a sum of digits in [1, 9]. In particular, E belongs to [1, 9].Being in [1, 9] and equivalent to 1 mod 9, E must be 1.
Heck!Where did I let my calculator!?
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8 comments:
Wrong puzzle !
Should be : 1999^1999
and also hint is A = B = C = D (mod 9)
btw the answer is 1
B=1+9+9+9+1+9+9+9=38
C=3+8=11
D=1+1=2
Hi all,
the question is 1999^1999
sorry for the mistake
Ok, that makes it a nicer altho' harder problem.
The series 1999^1, 1999^2, 1999^3, 1999^4 all have a digital root of 1; eg 1999^4==1 (Mod 9). I'm guessing that 3 cycles of extracting the digital root of the thousands of digits of 1999^1999 is sufficient to reach the digital root; ie 1.
I like your writing style. Nice blog.
Let E be the sum of digits of D.
Following the hint above (A, B, C, D and E are all equivalent mod 9) and using 1999 = 222*9 + 1 we get that E is equivalent to 1^1999 = 1 mod 9.
Let n(x) denote the number of digits of x. We have
n(A) <= 1999*log_{10}(1999) + 1 <= 1999*4 + 1 <= 8000.
Therefore, B <= 9*n(A) <= 72000 which implies n(B) <= 5.
On the same way we get:
C <= 9*n(B) <= 45,
n(C) <= 2, and
D <= 9*n(C) <= 18.
Every natural number in [1, 18] has a sum of digits in [1, 9]. In particular, E belongs to [1, 9].
Being in [1, 9] and equivalent to 1 mod 9, E must be 1.
Heck!
Where did I let my calculator!?
Post a Comment