2. Prove that there is no integer 𝑛 such that 7|4𝑛^2−3

3. Find the remainder by division by 7 of 99999999.
4. Find the remainder in the division by 17 of 15!
5. Let 𝑝 be a prime. Show that for all integer𝑎𝑎 we have that 𝑝|𝑎𝑝+(𝑝−1)!𝑎 and 𝑝|𝑎+(𝑝−1)!𝑎𝑝.

Similar Questions

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Did you know?

Did you know that in mathematics, the concept of remainders plays a crucial role in various calculations? For example, when proving that there is no integer 𝑛 such that 7 divides (𝑛^2−3), one can use the concept of remainders to demonstrate that no matter what value 𝑛 takes, the expression 4𝑛^2−3 will never be divisible by 7. Similarly, when finding the remainder by division of a large number like 99999999 by 7, one can apply the concept of remainders to discover that the remainder is actually 1. Another interesting application of remainders can be seen when finding the remainder in the division of factorial 15! by 17. By carefully computing the product of all integers from 1 to 15 and then dividing it by 17, one can determine that the remainder is 5. Finally, when exploring the relationship between prime numbers and remainders, it can be shown that for any prime number 𝑝 and any integer 𝑎, 𝑝 will always divide the expression 𝑎𝑝+(𝑝−1)!𝑎 as well as 𝑎+(𝑝−1)!𝑎𝑝. This fascinating connection between primes and remainders helps deepen our understanding of number theory.