Given:(1) If n is an even number, then 5n + 2 is always an even number.(2) If n is an integer larger than 1, then 2^n − 1 is always a prime number.Select one of the following choices:Question 16Answera.(1) and (2) are True.b.(1) is True and (2) is False.c.(1) and (2) are False.d.(1) is False and (2) is True.

For each of the following statements, either prove that the statement is true, or finda counterexample to show that the statement is false and explain your reasoning.(a) For each integer n, n is even if and only if 6n + 4 is even.(b) For all integers ℓ, m, n, if ℓ + m is odd and ℓ + n is even, then m + n is odd.

6. Use a direct proof to show that the sum of two odd integers is even.7. Use a direct proof to show that the sum of two even integers is even Show that if n is an integer andn3 + 5 is odd, then n is even usinga) a proof by contraposition.b) a proof by contradiction.8. Prove that if n is a positive integer, then n is odd if and only if 5n + 6 is odd.9. Find a counterexample to the statement that every positive integer can be written as the sum of thesquares of three integers.

There is an integer n > 5 such that 2^n − 1 is prime. Can you prove this?

Statement A (Assertion): 3" cannot end with digit zero for any natural number n Statement B (Reason): Any positive integer ending with the digit zero is divisible by 2 and 5 both. a. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) b. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) c. Assertion (A) is true but reason (R) is false d. Assertion (A) is false but reason (R) is true

Let P(x, y) is "x + 2y = xy," where x and y are integers.Given:(1) ForEvery x ThereExists y P(x, y).(2) ThereExists y ForEvery x P(x, y).Select one of the following choices:Question 5Answera.(1) and (2) are True.b.(1) and (2) are False.c.(1) is True and (2) is False.d.(1) is False and (2) is True.