using PMI show that ∀ n≥2 the number 5^n ends with the digits 25


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Question Number 33462 by NECx last updated on 17/Apr/18

$u s i n g P M I s h o w t h a t \forall n ⩾ 2 t h e$ $n u m b e r 5^{n} e n d s w i t h t h e d i g i t s 25$
	Answered by MJS last updated on 17/Apr/18

	$b a s e :$ $n = 2$ $5^{2} = 25$ $if true for n \Rightarrow true for n + 1$ $i, j \in N$ $5^{n} = 100 i + 25 \Rightarrow 5^{n + 1} = 100 j + 25$ $5^{n + 1} = (100 i + 25) \times 5 = 500 i + 125 =$ $= 500 i + 100 + 25 = 100 j + 25 with$ $j = 5 i + 1$ $this holds for n ⩾ 2$
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