Tessellate S.T.E.M.S - Mathematics - School - Set 2 - Problem 5

Number Theory Level 2

Find the last three digits of $\displaystyle 2^{2018}$ .

This problem is a part of Tessellate S.T.E.M.S.

624 544 224 144

1 solution

Patrick Corn
Jan 2, 2018

$2^{2018} \equiv 0$ mod $8$ and $2^{2018} \equiv 2^{18}$ mod $125$ since $\phi(125) = 100.$ Now $2^9 = 512 \equiv 12$ mod $125,$ so $2^{18} = (2^9)^2 \equiv 12^2 =144 \equiv 19$ mod $125.$

The congruences mod $8$ and mod $125$ imply that $2^{2018} \equiv \fbox{144}$ mod $1000.$

OK how we will solve if we have to find Last 4 digits of 7^128

Ayush Kumar - 3 years, 4 months ago

Tessellate S.T.E.M.S - Mathematics - School - Set 2 - Problem 5

1 solution

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