[Dana] Add factors and primeFactors programs (#58)

* Find prime Factors of a int and Find all non-trivial factors of an int Programs.

---------

Co-authored-by: Kostis Sagonas <kostis@cs.ntua.gr>

Author: JohnnyPol

dana/programs/factors.dana

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+(* 
+   This program reads a positive integer from the user and prints all its factors.
+   For the special cases where the number is 2 or 3, it directly prints the two factors: 1 and the number.
+*)
+
+def main
+  var n is int
+  var i is int
+
+  # Read the input
+  n := readInteger()
+
+  writeString: "Factors:"
+  
+  # If n is 2 or 3, print the factors 1 and n, then exit
+  if n = 2 or n = 3:
+    writeString: " 1 "
+    writeInteger: n
+    exit
+
+  # For all other numbers, iterate from 1 to n and print divisors
+  i := 2
+  loop:
+    if i > n / 2: break
+    elif n % i = 0:
+      writeString: " "
+      writeInteger: i  # i is a factor of n
+    i := i + 1

dana/programs/factors.input

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+12

dana/programs/factors.result

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+Factors: 2 3 4 6

dana/programs/primeFactors.dana

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+def main
+  # Helper procedure: prints the prime factors of n
+  def primeFactors: n as int
+    # Divide out all factors of 2
+    loop:
+      if n % 2 = 0:
+        writeString: " 2"
+        n := n / 2
+      else: break
+
+    var i is int
+    i := 3
+    # Loop over odd numbers starting from 3 while i*i <= n
+    loop:
+      if i * i <= n:
+        loop:
+          if n % i = 0:
+            writeString: " "
+            writeInteger: i
+            n := n / i
+          else: break
+        i := i + 2
+      else: break
+
+    # If remaining n is a prime greater than 2, print it
+    if n > 2:
+      writeString: " "
+      writeInteger: n
+
+  # Main program: Read input and call primeFactors
+  var num is int
+  num := readInteger()
+  writeString: "Prime factors:"
+  primeFactors: num
+  writeString: "\n"

dana/programs/primeFactors.input

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+153569623460

dana/programs/primeFactors.result

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+Prime factors: 2 2 5 11 223 3130241