Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

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

N=1335 in base a.   Write N in base a+1

$${N}=\mathrm{1335}\:{in}\:{base}\:{a}.\: \\ $$$${Write}\:{N}\:{in}\:{base}\:{a}+\mathrm{1} \\ $$

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

N=(1335)_a   =1×a^3 +3×a^2 +3×a+5  =1×a^3 +3×a^2 +3×a+1+4  =(a+1)^3 +4  =1×(a+1)^3 +0×(a+1)^2 +0×(a+1)+4  =(1004)_(a+1)

$${N}=\left(\mathrm{1335}\right)_{{a}} \\ $$$$=\mathrm{1}×{a}^{\mathrm{3}} +\mathrm{3}×{a}^{\mathrm{2}} +\mathrm{3}×{a}+\mathrm{5} \\ $$$$=\mathrm{1}×{a}^{\mathrm{3}} +\mathrm{3}×{a}^{\mathrm{2}} +\mathrm{3}×{a}+\mathrm{1}+\mathrm{4} \\ $$$$=\left({a}+\mathrm{1}\right)^{\mathrm{3}} +\mathrm{4} \\ $$$$=\mathrm{1}×\left({a}+\mathrm{1}\right)^{\mathrm{3}} +\mathrm{0}×\left({a}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{0}×\left({a}+\mathrm{1}\right)+\mathrm{4} \\ $$$$=\left(\mathrm{1004}\right)_{{a}+\mathrm{1}} \\ $$

Answered by MJS_new last updated on 18/Nov/20

1004  a^3 +3a^2 +3a+5=α(a+1)^3 +β(a+1)^2 +γ(a+1)+δ  (1−α)a^3 +(3−3α−β)a^2 +(3−3α+2β−γ)a+(5−α−β−γ−δ)=0  all brackets must be zero ⇒  α=1∧β=0∧γ=0∧δ=4

$$\mathrm{1004} \\ $$$${a}^{\mathrm{3}} +\mathrm{3}{a}^{\mathrm{2}} +\mathrm{3}{a}+\mathrm{5}=\alpha\left({a}+\mathrm{1}\right)^{\mathrm{3}} +\beta\left({a}+\mathrm{1}\right)^{\mathrm{2}} +\gamma\left({a}+\mathrm{1}\right)+\delta \\ $$$$\left(\mathrm{1}−\alpha\right){a}^{\mathrm{3}} +\left(\mathrm{3}−\mathrm{3}\alpha−\beta\right){a}^{\mathrm{2}} +\left(\mathrm{3}−\mathrm{3}\alpha+\mathrm{2}\beta−\gamma\right){a}+\left(\mathrm{5}−\alpha−\beta−\gamma−\delta\right)=\mathrm{0} \\ $$$$\mathrm{all}\:\mathrm{brackets}\:\mathrm{must}\:\mathrm{be}\:\mathrm{zero}\:\Rightarrow \\ $$$$\alpha=\mathrm{1}\wedge\beta=\mathrm{0}\wedge\gamma=\mathrm{0}\wedge\delta=\mathrm{4} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com