Answer to Question #26868 in Algebra for Prakkash Manohar

1. What is the 50th smallest positive integer that can be written as the sum of distinct powers of 3 with non-negative integer exponents?
2. How many positive integers n≤1000 cannot be written in the form a^2−b^2−c^2 where a,b and c are non-negative integers subject to a≥b+c?
3. Let 3^a be the highest power of 3 that divides 1000!. What is a?
4. We define n♡ recursively as follows.
1♡=1; n♡=((n−1)♡)⋅n+1
Find the largest n<1000 such that the last two digits of n♡ are zeroes.
5. How many integers appear in both of the following arithmetic progressions:
A1:2,9,16,..., 2+(1000−1)×7
A2:3,12,21,..., 3+(1000−1)×9