A Pretti number is a seven-digit positive integer with the following properties:
• The integer formed by its leftmost three digits is a perfect square.
• The integer formed by its rightmost four digits is a perfect cube.
• Its ten thousands digit and ones (units) digit are equal.
• Its thousands digit is not zero.
How many Pretti numbers are there?

How many 3-digit positive integers can be added to 463 such that no carry occurs when adding the digits?

9. Prove that there exists an integer a > 17 such that for every integer b, ab22 is a perfect square. [3 marks)

4. How many natural numbers between 100 and 999 have three distinct digits? (You do not need to obtain a single number. Leave your answer as a product of integers.)