Introduction

When dealing with large exponents in modular arithmetic, such as finding the remainder of 152010 * 162011 when divided by 7, it can be challenging. This process involves understanding fundamental principles in number theory. In this guide, we'll explore this problem using Fermat's Little Theorem and other concepts of modular arithmetic.

Understanding Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers where numbers ' wraps around' upon reaching a certain value, known as the modulus. For example, when dealing with numbers modulo 7, any number greater than or equal to 7 is reduced by the modulus until the result is within the range 0 to 6.

Step-by-step Solution

Step 1: Simplify (15^{2010} mod 7)

First, we simplify each term modulo 7:

15 equiv 1 mod 715^{2010} equiv 1^{2010} equiv 1 mod 7

Step 2: Simplify (16^{2011} mod 7)

For the second term, we note that:

16 equiv 2 mod 716^{2011} equiv 2^{2011} mod 7

Using Fermat's Little Theorem, we know:

2^{6} equiv 1 mod 7

We need to express (2^{2011}) in terms of (2^6):

2011 div 6  335 text{ remainder } 52^{2011} equiv 2^{5} mod 7

Now, calculate (2^5):

2^5  3232 mod 7  42^{2011} equiv 4 mod 7

Step 3: Combine the Results

Now, we combine the results from steps 1 and 2:

15^{2010} * 16^{2011} equiv 1 * 4 mod 71 * 4 mod 7  4

However, there was an error in the final step of combining the results. We should instead calculate:

1 * 2 mod 7  2

Thus, the correct remainder is 3:

15^{2010} * 16^{2011} equiv 1 * 2 equiv 2 mod 7

Alternative Approaches

Part One: Direct Simplification

We can break it into simpler steps:

15 equiv 1 mod 715^{2010} equiv 1^{2010} equiv 1 mod 7 16 equiv 2 mod 716^{2011} equiv 2 mod 7 (since 16^{3} equiv 2 mod 7) Combine these results:1 times 2 equiv 2 mod 7

Thus, the remainder is 2.

Part Two: Another Approach Using Modular Arithmetic

Using the properties of modular arithmetic and simplification:

15 equiv 1 pmod{7} implies15^{2010} equiv 1^{2010} equiv 1 pmod{7}16 equiv 2 pmod{7} implies16^{3} equiv 1 pmod{7}16^{2011}  16^{3 times 670} times 16 equiv 1^{670} times 16 equiv 16 equiv 2 pmod{7}

Therefore,15^{2010} times 16^{2011} equiv 1 times 2 equiv 2 pmod{7}

Thus, the remainder is 2.

Conclusion

In summary, the remainder when 152010 * 162011 is divided by 7 is 3. This was determined using different methods, all of which highlight the power of modular arithmetic in simplifying complex calculations involving large exponents.