Fermat's Little Theorem for `a=7` and `p=13`Solution:Fermat's Little Theorem
`a^(p-1)-=1 ("mod "p)` (where `p` is prime, `a` is not divisible by `p`)
Here `a=7, p=13` (Given)
`p=13` is prime. (First condition holds)
`a=7` is not divisible by `p=13`. (Second condition holds)
Now, we find remainder using Modulo method
`a^(p-1) " mod "p`
`7^12" mod "13`
Here `7^12=(7^2)^6`
`=(7^2" mod "13)^6" mod "13`
`=(49" mod "13)^6" mod "13`
`=10^6" mod "13`
Here `10^6=(10^2)^3`
`=(10^2" mod "13)^3" mod "13`
`=(100" mod "13)^3" mod "13`
`=9^3" mod "13`
Here `9^3=(9^2)^1*9`
`=((81" mod "13)*9)" mod "13`
`=(3*9)" mod "13`
`=27" mod "13`
`=1`
`:.` The remainder is `1`
So the values satisfy the Fermat's Little Theorem.
This material is intended as a summary. Use your textbook for detail explanation.
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