can you help me with whose problems in sagemath? thank you
Problem 1 Write a function that takes in an integer and determines whether it is a prime (i.e. has no divisors other than 1 and itself). There is a built in function is_prime that does this, but don't use it or any other built-in functions. Use your function to determine whether 2 - 1 is prime. Include in your Jupyter notebook a short written explanation of the algorithm you are using to test whether numbers are prime. Problem 2 Write a function that takes as input an integer n and returns its factorization into prime numbers. The output should be a list of the prime factors, with each prime p appearing k times in the list where k is the largest power of p such that pk divides n. The order of the elements of the list does not matter. For instance, if n = 12, any of (2,2,3), (2,3,2], or (3,2,2] would be vaild outputs. Do not use the built-in factor function. Test your function on 222 - 1,223 1, and 224 1. Problem 3 (a) Write a function that takes as input a string of characters, and returns the characters in reverse order (either as a list of characters, or as a string). Try your function on "MADWARWOLF". It should return "FLOWRAWDAM" (or a list of these characters). (b) Write a function that takes as input a string of characters, and applies the above reverse cipher, followed by a shift (Caesar) cipher like the one we did in class (shifting by an amount of your choice). How hard would it be to break this combination cipher? Problem 1 Write a function that takes in an integer and determines whether it is a prime (i.e. has no divisors other than 1 and itself). There is a built in function is_prime that does this, but don't use it or any other built-in functions. Use your function to determine whether 2 - 1 is prime. Include in your Jupyter notebook a short written explanation of the algorithm you are using to test whether numbers are prime. Problem 2 Write a function that takes as input an integer n and returns its factorization into prime numbers. The output should be a list of the prime factors, with each prime p appearing k times in the list where k is the largest power of p such that pk divides n. The order of the elements of the list does not matter. For instance, if n = 12, any of (2,2,3), (2,3,2], or (3,2,2] would be vaild outputs. Do not use the built-in factor function. Test your function on 222 - 1,223 1, and 224 1. Problem 3 (a) Write a function that takes as input a string of characters, and returns the characters in reverse order (either as a list of characters, or as a string). Try your function on "MADWARWOLF". It should return "FLOWRAWDAM" (or a list of these characters). (b) Write a function that takes as input a string of characters, and applies the above reverse cipher, followed by a shift (Caesar) cipher like the one we did in class (shifting by an amount of your choice). How hard would it be to break this combination cipher
