Find the last two digits of 7^(77).
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Step-by-Step Solution
Step 1: Understand the Problem
We need to find the last two digits of 777. This is equivalent to finding the value of 777 modulo 100. To do this, we will use Euler's totient theorem.
Step 2: Calculate Euler's Totient Function for 100
Euler's totient function, ϕ(n), counts the number of positive integers up to a given integer n that are relatively prime to n. For n=100, we find its prime factorization 100=22⋅52. Using the formula ϕ(n)=n∏p∣n(1−1/p), we calculate ϕ(100).
Step 3: Simplify the Exponent Modulo ϕ(100)
According to Euler's totient theorem, if a and n are coprime, then aϕ(n)≡1(modn). In our case, 7 and 100 are coprime. So, 777≡7(77(modϕ(100)))(mod100). We need to calculate the exponent 77 modulo ϕ(100)=40.
Step 4: Calculate 77(mod40)
We calculate the powers of 7 modulo 40 until we find a pattern or reach 77. We find that 74≡1(mod40). This simplifies the calculation for 77.
Step 5: Substitute and Calculate Final Result
Now we substitute the result from the previous step back into the original expression. We need to calculate 723(mod100). We can do this by breaking down the exponent and using modular arithmetic.
Step 6: Calculate 723(mod100)
We calculate powers of 7 modulo 100. We find that 74≡1(mod100). This is a very useful property. We can then write 723 as (74)5⋅73 and substitute the values.