The-unit-digit-of-the-square-of-a-number-and-the-units-digit-of-the-cube-of-the-number-are-equal-to-the-unit-digit-of-the-number-How-many-values-are-possible-for-the-units-digits-of-such-numbers-

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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

$$letabetheunitdigitofsuchnumbers.$$$$fromcondition1:$$$$a^2=10b+a$$$$\Rightarrow (a-1)a=10b$$$$i.e.thelastdigitoftheproductfrom$$$$(a-1)andamustbezero.$$$$\Rightarrow a=0,1,5,6$$$$$$$$fromcondition2:$$$$a^3=10b+a$$$$\Rightarrow (a-1)a(a+1)=10b$$$$i.e.thelastdigitoftheproductfrom$$$$(a-1),aand(a+1)mustbezero.$$$$\Rightarrow a=0,1,4,5,6,9$$$$$$$$\Rightarrow tofulfillbothconditions,$$$$a=0,1,5,6$$

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

$$wow$$

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