Assignment 01 Solution

  Requirements
- Write a C application with a menu based console interface which solves one of the problems below. 
- Menu entries are expected for *reading a vector of numbers from the console*, *solving each of the 2 required functionalities* and *exiting the program*. 
- Each requirement must be resolved using at least one function. Functions communicate via input/output parameters and the return statement.
- Provide specifications for all functions.\
**due in week 2.**

  Problem Statements
1.  **a.** Generate all the prime numbers smaller than a given natural number `n`.\
**b.** Given a vector of numbers, find the longest increasing contiguous subsequence, such the sum of that any 2 consecutive elements is a prime number.

2.  **a.** Generate the first `n` prime numbers (`n` is a given natural number).\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that any two consecutive elements are relatively prime.

3.    **a.** Print the Pascal triangle of dimension `n` of all combinations `C(m,k)` of m objects taken by `k, k = 0, 1, ..., m`, for line `m, where m = 1, 2, ..., n`.\
**b.** Given a vector of numbers, find the longest contiguous subsequence of prime numbers.

4.    **a.** Compute the approximated value of square root of a positive real number. The precision is provided by the user.\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that the difference of any two consecutive elements is a prime number.

5.    **a.** Print the exponent of a prime number `p` from the decomposition in prime factors of a given number `n` (n is a non-null natural number).\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that any two consecutive elements are relatively prime.

6.    **a.** Read a sequence of natural numbers (sequence ended by `0`) and determine the number of `0` digits of the product of the read numbers.\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that the sum of any two consecutive elements is a prime number.

7.    **a.** Read sequences of positive integer numbers (reading of each sequence ends by `0`, reading of all the sequences ends by `-1`) and determine the maximum element of each sequence and the maxim element of the global sequence.\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that all elements are in a given interval.

8.    **a.** Determine the value `x^n`, where `x` is a real number and `n` is a natural number, by using multiplication and squared operations.\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that any two consecutive elements have contrary signs.

9.    **a.** Decompose a given natural number in its prime factors.\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that any consecutive elements contain the same digits.

10.    **a.** Decompose a given even natural number, greater than 2, as a sum of two prime numbers (Goldbach’s conjecture).\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that any consecutive elements have at least 2 distinct digits in common.

11.    **a.** Determine the first `n` pairs of twin numbers, where n is a given natural and non-null number. Two prime numbers p and q are called twin if `q – p = 2`.\
**b.** Given a vector of numbers, find the longest decreasing contiguous subsequence.

12.    **a.** Determine all the numbers smaller than a given natural and non-null number `n` and that are relatively prime to n.\
**b.** Given a vector of numbers, find the longest contiguous subsequence with the maximum sum.

13.    **a.** Determine the first (and only) 8 natural numbers `(x1, x2,  …,  x8)` greater than 2 with the following property: all the natural numbers smaller than `xi` and that are relatively prime with `xi` (except for the number 1) are prime, `i =1,2, …, n`.\
**b.** Given a vector of numbers, find the longest contiguous subsequence such that any consecutive elements contain the same digits.