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

$The unit digit of the square of a number$ $and the units digit of the cube of the$ $number are equal to the unit digit of$ $the number . How many values are$ $possible for the units digits of such$ $numbers ?$
	Answered by mrW2 last updated on 07/Feb/18

	$l e t a b e t h e u n i t d i g i t o f s u c h n u m b e r s .$ $f r o m c o n d i t i o n 1 :$ $a^{2} = 10 b + a$ $\Rightarrow (a - 1) a = 10 b$ $i . e . t h e l a s t d i g i t o f t h e p r o d u c t f r o m$ $(a - 1) a n d a m u s t b e z e r o .$ $\Rightarrow a = 0, 1, 5, 6$ $f r o m c o n d i t i o n 2 :$ $a^{3} = 10 b + a$ $\Rightarrow (a - 1) a (a + 1) = 10 b$ $i . e . t h e l a s t d i g i t o f t h e p r o d u c t f r o m$ $(a - 1), a a n d (a + 1) m u s t b e z e r o .$ $\Rightarrow a = 0, 1, 4, 5, 6, 9$ $\Rightarrow t o f u l f i l l b o t h c o n d i t i o n s,$ $a = 0, 1, 5, 6$
	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

	$w o w$
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