1. Recall:
d is a divisor of n  n % d == 0
last digit of n  last = n % 10
removing last digit  n = n // 10
adding digit to n  n = n * 10 + digit
How to solve a problem?
identify inputs and outputs
find the calculations, selection and repetitions to solve it

2. A function is a piece of code introduced by def that:
Python functions
Python programs are made of a sequence of functions. These function can be called within the program or outside in the shell.
A function is a piece of code introduced by def that:
Has a name
Has some arguments which are the inputs
Returns some values which are the outputs
Has some code to calculate the outputs from the inputs

3. The function head starts with def and finished with :
def functionName(arg1, arg2, …):
statem1
statem2 …
return output
# end functionName
The function head starts with def and finished with :
The function body calculates the output from the input args
The output is returned.
The function is called by its name
output = functionName(a1,a2,..)

4. Function to calculate the min, max values of a and b.
- inputs: a, b are arguments
- output: min, max must returned
- how to calculate? Use an if selection

5. def minMax(a, b) :
‘’’ function to calculate the max and min values of a and b. ‘’’
if a>b :
max, min = a, b
else:
max, min = b, a
# endif
return min, max
# end minMax

6. ‘’’ find the max value for a,b,c,d.’’’ min1, max1 = minMax(a,b)
Function to calculate the maximum value of 4 numbers a, b, c, d.
def maximum4(a,b,c,d) :
‘’’ find the max value for a,b,c,d.’’’
min1, max1 = minMax(a,b)
min2, max2 = minMax(c,d)
min, max = minMax(max1, max2)
return max
# end maximum4

7. Functions are written once and used multiple times.

8. Prime numbers
n is prime when:
1) it has only 2 divisors 1 and n
2) it has not any proper divisor in range of
2, 3, 4,…, sqrt(n)

9. Write a function to test if a number n is prime.
- input: n
- output: ans
- how to do it?
1) test 2  ans = True
3) test even number  ans = False
4) repeat for a potential odd divisor 3,5,.., ?
if d is a divisor then ans = False
ans = true

10. def isPrime(n):
‘’’ function to test primality.’’’
if n == 1 :
return False
# endif
if n == 2 :
return True
#endif
for d in range(3, math.sqrt(n)+1;2) :
if n % d == 0 :
return False
# endif
#endfor
return True

11. The Euclidian Algorithm Find the gcd of 2 numbers a, b. Solutions:
Find prime decomposition and choose the common primes with the smallest exponents
a = 2**4 * 3**2 * 5 and b = 2 * 5 **2 gcd = 2 ** 1 * 5 **1
Euclidian algorithm does consecutive divisions
Divide a to b to find res and rem
Change a = b and b = rem

12. Example a = 56 and b = 40
1) a = 56, b= 40  res = a // b = 1 rem = a % b = 16
2) a = 40, b = 16  res = a // b = 2 rem = a%b =8
3) a = 16, b = 8 res = a//b = 2 rem = a%b =0
4) a = 8, b = 0  repetition stops
gcd = 8 the last remainder different than 0

13. The number of division in the algorithm is log (max(a,b)) hence just few divisions.