Number systems

Let $AP(a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If

$$AP(1;3)\cap AP(2;5)\cap AP(3;7) = AP(a;d)$$

then $a + d$ equals ______

If $a_4 = 28$, then $p + 2q =$

If $(27)^{999}$ is divided by 7, then the remainder is :

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is :

A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k-20=$

Let $z=x+iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation
$$z\bar z^3+\bar zz^3=350$$
is