use fermat's theorem to find a number x between 0 and 28 with x85 congruent to 6 modulo 29
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Find x between 0 and 28 with x^85 ≡ 6 mod (35) using FLT
“Fermat's theorem” might refer to one of the several theorems, but the most likely choice in this context is Fermat's little theorem: If p is a prime number, then for any integer a, the number a^p − a is an integer multiple of p, or
a^p ≡ a mod p
6 mod (35) ≡ 6 mod (5) ≡ 1 mod (5)
To see the logic explaining why this is true, consider this example
111 = 6 + 3*35 which is 6 mod (35)
35 = 7*5, so 111 = 6 + 21*5 which is 6 mod (5)
That example demonstrates that we may always write
6 mod (35) ≡ 6 mod (5) ≡ 1 mod (5) .........(1)
a^5 ≡ a mod 5 by FLT, so raising both sides to 17th power
a^85 ≡ (a^17) mod 5 ............................(2)
a^85 ≡ 6 mod (35) we must select one of the values of a that satisfy
(a^17) ≡ 1 mod (5)
This works for values of a given by a = 5n + 1 and between 0 and 28 the
options are:- 6, 11, 16, 21, 26
“Fermat's theorem” might refer to one of the several theorems, but the most likely choice in this context is Fermat's little theorem: If p is a prime number, then for any integer a, the number a^p − a is an integer multiple of p, or
a^p ≡ a mod p
6 mod (35) ≡ 6 mod (5) ≡ 1 mod (5)
To see the logic explaining why this is true, consider this example
111 = 6 + 3*35 which is 6 mod (35)
35 = 7*5, so 111 = 6 + 21*5 which is 6 mod (5)
That example demonstrates that we may always write
6 mod (35) ≡ 6 mod (5) ≡ 1 mod (5) .........(1)
a^5 ≡ a mod 5 by FLT, so raising both sides to 17th power
a^85 ≡ (a^17) mod 5 ............................(2)
a^85 ≡ 6 mod (35) we must select one of the values of a that satisfy
(a^17) ≡ 1 mod (5)
This works for values of a given by a = 5n + 1 and between 0 and 28 the
options are:- 6, 11, 16, 21, 26
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