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Number Theory Level 2

( T H I R T Y ) 35 + ( F I V E ) 35 (THIRTY)_{35}+(FIVE)_{35}

Find the sum of the above two base-35 numbers in base 35.

Clarifications:

For base-35 numbers: 0 = 0 0=0 , 1 = 1 1=1 , 2 = 2 2=2 , ... 9 = 9 9=9 , A = 10 A=10 , B = 11 B=11 , C = 12 C=12 , ... X = 33 X=33 , Y = 34 Y=34 , and Z = 35 Z=35 .

For the answer, enter the sum of the decimal values of the base-35 digits. For example:

  • If the sum is R E A L L Y REALLY and R E A L L Y = 27 + 14 + 10 + 21 + 21 + 34 = 127 REALLY=27+14+10+21+21+34=127 , then the answer is 127 127 .
  • If the sum is 45 R E A L L Y = 4 + 5 + 27 + 14 + 10 + 21 + 21 + 34 = 136 45REALLY=4+5+27+14+10+21+21+34=136 , then the answer is 136 136 .


The answer is 130.

Abhay Tiwari by Abhay Tiwari , 28, India

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1 solution

S = ( T H I R T Y ) 35 + ( F I V E ) 35 = 3 5 5 T 35 + 3 5 4 H 35 + 3 5 3 I 35 + 3 5 2 R 35 + 35 T 35 + Y 35 + 3 5 3 F 35 + 3 5 2 I 35 + 35 V 35 + E 35 = 3 5 5 ( 29 ) + 3 5 4 ( 17 ) + 3 5 3 ( 18 ) + 3 5 2 ( 27 ) + 35 ( 29 ) + 34 + 3 5 3 ( 15 ) + 3 5 2 ( 18 ) + 35 ( 31 ) + 14 = 3 5 5 ( 29 ) + 3 5 4 ( 17 ) + 3 5 3 ( 18 + 15 ) + 3 5 2 ( 27 + 18 ) + 35 ( 29 + 31 ) + 34 + 14 = 3 5 5 ( 29 ) + 3 5 4 ( 17 ) + 3 5 3 ( 33 ) + 3 5 2 ( 45 ) + 35 ( 60 ) + 48 = 3 5 5 ( 29 ) + 3 5 4 ( 17 ) + 3 5 3 ( 33 ) + 3 5 2 ( 45 ) + 35 ( 60 + 1 ) + 13 = 3 5 5 ( 29 ) + 3 5 4 ( 17 ) + 3 5 3 ( 33 ) + 3 5 2 ( 45 + 1 ) + 35 ( 26 ) + 13 = 3 5 5 ( 29 ) + 3 5 4 ( 17 ) + 3 5 3 ( 33 + 1 ) + 3 5 2 ( 11 ) + 35 ( 26 ) + 13 = 3 5 5 ( 29 ) + 3 5 4 ( 17 ) + 3 5 3 ( 34 ) + 3 5 2 ( 11 ) + 35 ( 26 ) + 13 \begin{aligned} S & = (THIRTY)_{35} + (FIVE)_{35} \\ & = 35^5T_{35} + 35^4H_{35} + 35^3I_{35} + 35^2R_{35} + 35T_{35} + Y_{35} + 35^3F_{35} + 35^2I_{35} + 35V_{35} + E_{35} \\ & = 35^5(29) + 35^4(17) + 35^3(18) + 35^2(27) + 35(29) + 34 + 35^3(15) + 35^2(18) + 35(31) + 14 \\ & = 35^5(29) + 35^4(17) + 35^3(18+15) + 35^2(27+18) + 35(29+31) + 34+14 \\ & = 35^5(29) + 35^4(17) + 35^3(33) + 35^2(45) + 35(60) + 48 \\ & = 35^5(29) + 35^4(17) + 35^3(33) + 35^2(45) + 35(60\color{#D61F06}{+1}) + \color{#D61F06}{13} \\ & = 35^5(29) + 35^4(17) + 35^3(33) + 35^2(45\color{#D61F06}{+1}) + 35(\color{#D61F06}{26}) + 13 \\ & = 35^5(29) + 35^4(17) + 35^3(33\color{#D61F06}{+1}) + 35^2(\color{#D61F06}{11}) + 35(26) + 13 \\ & = 35^5(\color{#3D99F6}{29}) + 35^4(\color{#3D99F6}{17}) + 35^3(\color{#3D99F6}{34}) + 35^2(\color{#3D99F6}{11}) + 35(\color{#3D99F6}{26}) + \color{#3D99F6}{13} \end{aligned}

Therefore, the answer is 29 + 17 + 34 + 11 + 26 + 13 = 130 \color{#3D99F6}{29+17+34+11+26+13} = \boxed{130} .

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Ah, Nice solution sir (+1). :)

Abhay Tiwari - 5 years ago

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