Performing calculations with updated integer
How about something like this?

import time

def add_num(x):
    added_int = int(raw_input("What number would you like to add to your integer (%s) by?" % x))
    outcome = x + added_int
    print("\nThe sum of %s + %s is %s" % (x, added_int, outcome))
    return outcome

def sub_num(x):
    sub_int = int(raw_input("What number would you like to subtract to your integer (%s) by?" % x))
    outcome = x - sub_int
    print("\nThe subtract of %s - %s is %s" % (x, added_int, outcome))
    return outcome

def mul_num(x):
    mul_int = int(raw_input("What number would you like to multiply your integer (%s) by?" % x))
    outcome = x * mul_int
    print("\nThe multiplication of %s * %s is %s" % (x, added_int, outcome))
    return outcome

def div_num(x):
    div_num = int(raw_input("What number would you like to divide your integer (%s) by?" % x))
    outcome = x / float(div_num)
    print("\nThe divider of %s / %s is %s" % (x, added_int, outcome))
    return outcome

def main():
    op_map = {"1":add_num, "2":sub_num, "3":mul_num, "4":div_num}
    number = 0
    print("The current integer is set at %s ." % number)
    while True:
        prompt = raw_input("Would you like to change the number? (Y/N)").lower()
        if prompt == "y":
            number = int(raw_input("Insert the new integer here: "))
            print("\nYou have changed the integer to %s .\n\n" % number)
            time.sleep(1)
            print("1. Add / 2. Subtract / 3. Multiply / 4. Divide\n")

            op = raw_input("What would you like to do with your new integer? (Choose a number) \n")
            if op is not None:
                operation = op_map.get(op)
                number = operation(number)

        elif prompt == "n":
            print("The integer will stay the same.")
            time.sleep(2)
            raw_input("Press enter key to exit...")
            break
        else:
            print("Invalid response")

if __name__ == '__main__':
    main()