Thursday, 2 November 2017

elementary number theory - Working out modulo without a calculator



How does one determine modulo without a calculator in cases like this:
15^7 - 13^5(\mod14)

Normally I would simply divide what is given by the modulo number and take the decimal output and times it by the modulo number. How can I work out 15^7 - 13^5(mod14) without the use of a calculator?



Now what I am thinking is:
15 \cong 1 \mod 14
15^7 \cong 1 \mod 14
13 \cong -1 \mod 14
13^2 \cong 1 \mod 14
13^5 \cong -1 \mod 14
[15^7 - 13^5(\mod 14)] = 1 (\mod 14) + 1 (\mod 14) = 2\mod 14
Is that right?



Answer



Yes, that's correct! [15^7 - 13^5]\pmod{14} \;\;= \;\;2\pmod {14}


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