Introduction to Fermat's Little Theorem
Before delving into the main content, it is crucial to understand the fundamental theorem we will be utilizing - Fermat's Little Theorem. This theorem is a cornerstone in number theory and is used extensively in solving problems related to remainders and modular arithmetic. It states that if (p) is a prime number and (a) is an integer not divisible by (p), then:
This theorem simplifies the calculation of large exponentiations modulo a prime number. Let's see how to use this theorem to find the remainder of (67^{99}) divided by 7.
Finding the Remainder of (67^{99}) Divided by 7
We begin the process by reducing the base 67 modulo 7. To do so, we perform the division:
[67 div 7 9 text{ with a remainder of } 4]Hence,
[67 equiv 4 text{ (mod } 7text{)}]Our goal now is to find (67^{99} mod 7). By the reduction performed, we need to calculate (4^{99} mod 7).
Using Fermat's Little Theorem
Since 7 is a prime number and 4 is not divisible by 7, we apply Fermat's Little Theorem:
[4^{6} equiv 1 text{ (mod } 7text{)}]Next, we need to reduce the exponent 99 modulo 6 (since 6 7-1):
[99 div 6 16 text{ with a remainder of } 3]Hence,
[99 equiv 3 text{ (mod } 6text{)}]So we have:
[4^{99} equiv 4^{3} text{ (mod } 7text{)}]Calculating the Result
Now, we calculate (4^{3}):
[4^{3} 64]Next, we find 64 mod 7:
[64 div 7 9 text{ with a remainder of } 1]Hence,
[64 equiv 1 text{ (mod } 7text{)}]Therefore, we can conclude that:
[67^{99} equiv 4^{99} equiv 4^{3} equiv 64 equiv 1 mod 7]The remainder of (67^{99}) when divided by 7 is 1.
Alternative Methods and Insights
Several alternative methods can be used to verify our result or attempt similar problems.
Method 1: Modular Arithmetic Reduction
We can also express 67 4 text{ (mod } 7text{)}end{p>
since 67 equiv 4 text{ (mod } 7text{)}end{p>
and by Fermat's Little Theorem, we know:
[4^{6} equiv 1 text{ (mod } 7text{)}]Then,
[4^{99} 4^{3 cdot 33} equiv (4^{6})^{16} cdot 4^{3} equiv 1^{16} cdot 4^{3} equiv 4^{3} text{ (mod } 7text{)}]Calculating 4^{3} end{p>
[4^{3} 64]
and
[64 mod 7 1]So,
[67^{99} equiv 1 text{ (mod } 7text{)}]Method 2: Using a More Direct Approach
Another way to look at this problem is to recognize some patterns in powers of 4 modulo 7:
[67 equiv 4 text{ (mod } 7text{)}]Thus,
[67^{99} equiv 4^{99} text{ (mod } 7text{)}]From the theorem, we know:
[4^2 equiv 2 text{ (mod } 7text{)}]and consequently,
[4^3 equiv 4^2 cdot 4 equiv 2 cdot 4 equiv 8 equiv 1 mod 7]Therefore,
[4^{99} equiv 1 text{ (mod } 7text{)}]So, the remainder of 67^{99} when divided by 7 is 1.
Conclusion
Through the application of Fermat's Little Theorem, we were able to efficiently determine the remainder of 67^{99} when divided by 7. This problem showcases the power of modular arithmetic and the utility of Fermat's Little Theorem in simplifying complex calculations.