What are the reminder if 8 power 25 is divided by 7 then?
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The best way to find remainder is by using Fermat’s Little Theorem which suggests:
If p is a prime and a is any integer not divisible by p, then {a^(p − 1)}− 1 is divisible by p.
8 is not a prime number.
Let us check factors of 8 which are 1, 2, 4, 8. We found a prime number i.e. 2.
So 2^6 mod 7 = 1
Thus, we can say 2 raised to multiples of power 6 mod 7 will also give remainder as 1.
Also 8 = 2^3
Hence 8^25 = 2^75
Finally,
2^75 mod 7
= 2^{(6 x 12) + 3} mod 7
= [1] x 2^3 mod 7
= 8 mod 7
= 1
Your final answer is 1.
If p is a prime and a is any integer not divisible by p, then {a^(p − 1)}− 1 is divisible by p.
8 is not a prime number.
Let us check factors of 8 which are 1, 2, 4, 8. We found a prime number i.e. 2.
So 2^6 mod 7 = 1
Thus, we can say 2 raised to multiples of power 6 mod 7 will also give remainder as 1.
Also 8 = 2^3
Hence 8^25 = 2^75
Finally,
2^75 mod 7
= 2^{(6 x 12) + 3} mod 7
= [1] x 2^3 mod 7
= 8 mod 7
= 1
Your final answer is 1.
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