Find-the-last-2-digits-from-20-17-17-20-

Question Number 23612 by Joel577 last updated on 02/Nov/17

Find the last 2 digits from

20^{17} + 17^{20}

Commented by Rasheed.Sindhi last updated on 02/Nov/17

20^{17} = {(2 \times 10)}^{17} = 2^{17} \times 10^{17}

∴ 20^{17} has 17 0^{'} s at the end .

∴ 20^{17} has no effect on last two

digits .

Hence we have to find last 2 digits

of 17^{20} only .

17^{1} = 17

17^{2} = 289

⋮

Commented by prakash jain last updated on 02/Nov/17

17^{2} \mod 100 = 80 = - 11

17^{20} \mod 100 = 11^{10}

11 \times 11 = 121

11^{2} = 21 (\mod 100)

21^{2} = 441 = 41 (\mod 100)

11^{10} = 21^{5} = 21^{4} \times 21 = {(41)}^{2} \times 21 (\mod 100)

{(41)}^{2} \times 21 = {(50 - 9)}^{2} \times 21

= 81 \times 21

= (- 19) \times 21

= - 399 = - 99 = 1 (\mod 100)