Answer :
Final answer:
To find the remainder of 5⁶⁸ divided by 196, we break down the problem into two steps. First, find the remainder when 5⁶⁸ is divided by 49, and then find the remainder when that result is divided by 4. The remainder of 5⁶⁸ divided by 196 is 1.
Explanation:
To calculate the remainder of 5⁶⁸ when divided by 196, we can use modular arithmetic. Since 196 is equal to 49 x 4, we can break down the calculation into two steps. First, find the remainder when 5⁶⁸ is divided by 49. We can do this by observing the pattern of remainders when powers of 5 are divided by 49:
5² = 25 (remainder 25)
5³ = 125 (remainder 27)
5⁴ = 625 (remainder 44)
5⁵ = 3125 (remainder 1)
5⁶ = 15625 (remainder 25)
5⁷ = 78125 (remainder 27)
5⁸ = 390625 (remainder 44)
...
We notice that the remainders repeat in the pattern 25, 27, 44, 1. So the remainder when 5⁶⁸ is divided by 49 is the same as the remainder when 5⁴ (the fourth element in the pattern) is divided by 49, which is 1.
Next, we find the remainder when 1 is divided by 4. Since 1 is less than 4, the remainder is 1.
Therefore, the remainder of 5⁶⁸ when divided by 196 is 1.
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Final answer:
The remainder of 5⁶⁸ when divided by 196 can be found using Euler's Theorem and modular arithmetic, which gives a result of 5^68.
Explanation:
To answer this question, we need to make use of a mathematical theorem called the 'Euler's Theorem', which is a fundamental tool in number theory. This theorem basically says that if 'a' and 'n' are coprime (have no common factors other than 1), then a^(phi(n)) ≡ 1 (mod n). Here, φ(n) is a function that gives the count of numbers from 1 to n that are coprime with n.
In this case, 'a' is 5, and 'n' is 196. We first need to calculate the 'Euler's totient' function for 196, denoted by φ(196). This is equal to 196 * (1 - 1/2) * (1 - 1/7) = 84.
From Euler's theorem, we know that 5^84 ≡ 1 (mod 196). This means that dividing any multiple of 84 by 196 will always leave a remainder of 1. Notice that 68 is less than 84. So, the remainder is just the same as 5^68.
In conclusion, by using the mathematical principles embodied in the Euler's Theorem, combined with the basic concepts of modular arithmetic, we are able to calculate the remainder of 5⁶⁸ when divided by 196, which is equal to 5^68.
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