Question Number 29326 by abdus salam last updated on 07/Feb/18
$$\mathrm{The}\:\mathrm{unit}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\:\mathrm{a}\:\mathrm{number} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{units}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{number}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{unit}\:\mathrm{digit}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{number}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{values}\:\mathrm{are}\: \\ $$$$\mathrm{possible}\:\mathrm{for}\:\mathrm{the}\:\mathrm{units}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{such} \\ $$$$\mathrm{numbers}? \\ $$
Answered by mrW2 last updated on 07/Feb/18
$${let}\:{a}\:{be}\:{the}\:{unit}\:{digit}\:{of}\:{such}\:{numbers}. \\ $$$${from}\:{condition}\:\mathrm{1}: \\ $$$${a}^{\mathrm{2}} =\mathrm{10}{b}+{a} \\ $$$$\Rightarrow\left({a}−\mathrm{1}\right){a}=\mathrm{10}{b} \\ $$$${i}.{e}.\:{the}\:{last}\:{digit}\:{of}\:{the}\:{product}\:{from} \\ $$$$\left({a}−\mathrm{1}\right)\:{and}\:{a}\:{must}\:{be}\:{zero}. \\ $$$$\Rightarrow{a}=\mathrm{0},\mathrm{1},\mathrm{5},\mathrm{6} \\ $$$$ \\ $$$${from}\:{condition}\:\mathrm{2}: \\ $$$${a}^{\mathrm{3}} =\mathrm{10}{b}+{a} \\ $$$$\Rightarrow\left({a}−\mathrm{1}\right){a}\left({a}+\mathrm{1}\right)=\mathrm{10}{b} \\ $$$${i}.{e}.\:{the}\:{last}\:{digit}\:{of}\:{the}\:{product}\:{from} \\ $$$$\left({a}−\mathrm{1}\right),{a}\:{and}\:\left({a}+\mathrm{1}\right)\:{must}\:{be}\:{zero}. \\ $$$$\Rightarrow{a}=\mathrm{0},\mathrm{1},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{9} \\ $$$$ \\ $$$$\Rightarrow{to}\:{fulfill}\:{both}\:{conditions}, \\ $$$${a}=\mathrm{0},\mathrm{1},\mathrm{5},\mathrm{6} \\ $$
Commented by Rasheed.Sindhi last updated on 07/Feb/18
3x(3113n7 5!r!
Commented by mrW2 last updated on 07/Feb/18
Now I understand. Thank you sir!
Commented by Rasheed.Sindhi last updated on 07/Feb/18
5up3r8!!!
Commented by NECx last updated on 08/Feb/18
$${wow} \\ $$