I came across a question which uses Fermat's little theorem:
The Question:
Find a positive integer n such that - 10 is divisible by 83.
The Solution given in the book:
Since 7 x 37 = 259 = 10 mod 83
We have to find a value of n such that = 7 x 37 mod 83
This is equivalent to = 37 = mod 83
By Fermat's Theorem,
= 1 mod 83 for all k.
So we need to choose n such that = mod 83
This will be satisfied if k=15
Therefore = mod 83
And so n =
This gives one value.
My problem is that I don't understand the fourth line
That is how 37 = mod 83?
So can someone please explain this?
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Top Newest28≡256≡249+7≡7 (mod 83)
⇒216≡72 (mod 83)
⇒220≡49⋅16≡784≡37(mod 83)
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Wow !! I didn't think of that. Thanks a LOT Sir !!!
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Note that 2 is a primitive root modulo 83, so that even if the remainder weren't 37, we could find another exponent (but finding the exact exponent is difficult, in general.) instead of 20 here.
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