integers a, b, and c with a≠0, if a∣b,  then a∣(bc). I need a know show table and a proof

For each of the following problems, you should write your answer as an expression. Do not give the final numeric value. For example, if your answer is the number of 6-permutations of a set with 10 elements, you should write P(10,6) or 10!/4! or 10 × 9 × 8 × 7 × 6 × 5, but not 151200. 5. How many different bit strings of length 9... 5.1 ...contain at least three O's? 5.2 ...start and end with the same digit and contain at most two 1's? 5.3 ...start with a 1 or contain exactly four 0's? 5.4...contain at least four consecutive 1's?

Find (with proof) all the pairs p, q of positive prime numbers that satisfy p? – 2q? = 1.

Give a direct proof: Let a, b, and c be integers. If a | b and a c, then a | (b. c). Remember that you must use the definition of | in your proof. Suppose a, b, and care integers and a | b and a | c. By the definition of 1, b = ak₁ and c = for some integers k₁ and k₂. Therefore, b . c = a, and is an integer, so a | b. c.

Consider all positive integers with seven different digits. (Note that zero cannot be the first digit, and by different, we mean that there is no repeated digit in the number.) b.) Find the number of them which is greater than 8000000.   c.) Find the number of them which are odd.   d.) Find the number of them which are divisible by 5.