N-1335-in-base-a-Write-N-in-base-a-1-

Question Number 122679 by mathocean1 last updated on 18/Nov/20

N = 1335 i n b a s e a .

W r i t e N i n b a s e a + 1

Answered by mr W last updated on 18/Nov/20

N = {(1335)}_{a}

= 1 \times a^{3} + 3 \times a^{2} + 3 \times a + 5

= 1 \times a^{3} + 3 \times a^{2} + 3 \times a + 1 + 4

= {(a + 1)}^{3} + 4

= 1 \times {(a + 1)}^{3} + 0 \times {(a + 1)}^{2} + 0 \times (a + 1) + 4

= {(1004)}_{a + 1}

Answered by MJS_new last updated on 18/Nov/20

1004

a^{3} + 3 a^{2} + 3 a + 5 = α {(a + 1)}^{3} + β {(a + 1)}^{2} + γ (a + 1) + δ

(1 - α) a^{3} + (3 - 3 α - β) a^{2} + (3 - 3 α + 2 β - γ) a + (5 - α - β - γ - δ) = 0

all brackets must be zero \Rightarrow

α = 1 \land β = 0 \land γ = 0 \land δ = 4

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