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# what is the ones digit of 9^40?

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what is the ones digit of 9^40?

Guest Feb 23, 2015

#2
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what is the ones digit of 9^40 ?

$$9^{40}\mod 10\\ =( \underbrace{9^2}_{=1\mod 10} )^{20}\mod 10 \qquad | \qquad 9^2\mod 10 =\ 81 \mod 10 =\ 1 \\\\ =1^{20}\mod 10 \qquad\qquad | \quad 1^{20} = 1 \\ =1\mod 10$$

heureka  Feb 24, 2015
#1
+91027
+10

Note the cycle

9^1 = 9

9^2 = 81

9^3 = 729

9^4 =  6561     and this pattern repeats for every "block" of 4 powers

Note that 40 is evenly divisible by 4, so 9^40 has a "ones" digit that completes this cycle. So....9^40  ends in a "1"

CPhill  Feb 23, 2015
#2
+20146
+10
$$9^{40}\mod 10\\ =( \underbrace{9^2}_{=1\mod 10} )^{20}\mod 10 \qquad | \qquad 9^2\mod 10 =\ 81 \mod 10 =\ 1 \\\\ =1^{20}\mod 10 \qquad\qquad | \quad 1^{20} = 1 \\ =1\mod 10$$