Skipping Bases

Number Theory Level 1

12 3 4 12 3 5 12 3 6 \large 123_4 \qquad 123_5 \qquad 123_6

The above shows three numbers, each written in a different base representation . Which of these numbers has the largest value?

12 3 5 123_5 12 3 6 123_6 12 3 4 123_4

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

12 3 6 = ( 6 2 × 1 ) + ( 6 1 × 2 ) + ( 6 0 × 3 ) = 51 123_6=(6^2×1)+(6^1×2)+(6^0×3)=51

12 3 5 = ( 5 2 × 1 ) + ( 5 1 × 2 ) + ( 5 0 × 3 ) = 38 123_5=(5^2×1)+(5^1×2)+(5^0×3)=38

12 3 4 = ( 4 2 × 1 ) + ( 4 1 × 2 ) + ( 4 0 × 3 ) = 27 123_4=(4^2×1)+(4^1×2)+(4^0×3)=27 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

It's easier to show that 1 n 2 + 2 n + 3 1n^2 + 2n+3 is an increasing function for n ≥ 4 n\geq 4 .

Pi Han Goh - 5 years, 1 month ago

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