2-digit numbers in base-10 which are divisible by their digits are
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99
http://oeis.org/A034838
2-digit numbers which are divisible by the sum of their digits are called Harshad numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90
http://oeis.org/A005349
2-digit numbers which are divisible by both their digits and the sum of their digits are
1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 48
http://oeis.org/A050104
Numbers which are equal to (i.e., not just divisible by) the product of their divisors and the sum of their divisors are called sum-product numbers and are given by
0, 1, 135, 144
http://oeis.org/A038369
(1) Find all 2-digit numbers that are divisible by the product of their digits
(2) Find all 2-digit numbers that have two distinct nonzero digits, such that the difference between the number and its reversal (the same number with the order of the digits reversed) is divisible by the sum of the digits of the number.
(3) Find all 2-digit numbers that, when added to their reversals, give perfect squares.
(4) Find all 2-digit numbers that, when subtracted from their reversals, give perfect squares.
(5) Find all 2-digit numbers that, when subtracted from their reversals, give perfect cubes.
(6) Find all 2-digit numbers with nonzero digits, that, when added to their reversals, give a result divisible by 66.
(7) Find two 2-digit numbers with distinct digits such that the difference between the square of the number and the square of the reversal of the number is itself a square.
