Difference-of-last-two-digits-of-2019-2019-2019-is-

Question Number 41720 by Tawa1 last updated on 11/Aug/18

Difference of last two digits of 2019^{2019^{2019}} is ?

Commented by MJS last updated on 12/Aug/18

= 2019 \times 2019 \times 2019 \times \dots

so we first need to know the sequence of the

last 2 digits of 19^{n} :

{19, 61, 59, 21, 99, 81, 39, 41, 79, 01, \dots}

the difference between these last 2 digits :

{8, 5, 4, 1, 0, 7, 6, 3, 2, 1, \dots}

both have a period of 10 \Rightarrow

we need to know the remainder of \frac{2019^{2019}}{10}

the sequence of the remainders of \frac{2019^{n}}{10}

{9, 1, 9, 1, \dots}

so the remainder is 9 if n is uneven and

1 if n is even

2019 is uneven \Rightarrow remainder is 9 \Rightarrow last two

digits of 2019^{2019^{2019}} = 79 \Rightarrow difference is 2

Commented by Tawa1 last updated on 15/Aug/18

God bless you sir

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