These questions are to be solved without a calculator:
1) Three different non-zero digits are used to form six different 3 digit numbers. The sum of 5 of them is 3321. What is the 6th number?
2) How many pairs \((a, b)\) of positive integers are there such that \(a\) and \(b\) are factors of \(6^6\) and \(a\) is a factor of \(b\)?
3) All digits of the positive integer \(N \)are either \(0\) or \(1\). The remainder after dividing\( N\) by \(37\) is \(18\). What is the smallest number of times that the digit \(1\) can appear in \(N\)?
4) In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged in a row so that the product of any 2 adjacent numbers is even? Choices are: 64 or 72 or 120 or 144 or 720
5) A hockey game between two teams is 'relatively close' if the number of goals scored by the two teams differs by more than two. In how many can the first 12 goals of a game is scored if the game is "relatively close"?
6) The 4 digit number \(pqrs\) has the property \(pqrs \times srqp\). If \(p = 2\) what is the value of the 3 digit number \(qrs\)?
7) If \(x^2 = x + 3\), then what is the value of \(x^3\)? Choices are: A) x + 6 B) 2x + 6 C) 3x + 9 D) 4x + 3 E) 27x + 9
Please answer these questions with complete steps in the comment section below. I thank everyone who tries from the bottom of my heart.