Prove that if the sum of S = a1 + a2 + … + an is an even integer, and all the terms a1, a2, …, are odd integers, then n is an even integer.

In each of the following statements, n represents a positive integer. One of the statements is true and the other two are false.  If n is even, then n is not prime  if n is a square number, then n is not prime  if n is a prime number and n ≥ 10, then n+2 is prime or n+4 is prime.  A. What two statements are false and in each case give a counter example to show that it is false  B. Prove the true statement

Assuming every integer is either even or odd, How would I show that for any integer, the difference between it and its cube is even

Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

2 Prove that a positive integer n is even if and only if nis even.

24. Suppose that I choose six of the first ten positive integers. Prove that I must have chosen two numbers such that one is a divisor of the other. [HINT: Write each of the 10 numbers as a power of 2 multiplied by an odd number, then use the pigeonhole principle.]

Prove the following statements: (a) If n is a positive integer then n 2 + 1 is an odd positive integer. (b) If n 2 + 1 is an even positive integer, and n is a positive integer then n is an odd positive integer.

Pls. answer these two questions by typing and not by handrwriting

ERCISES 5.1.7. Solve the following problems. 1) Find the number of 5-lists of the form (21, 22, 23, 24, 25), where each , is a nonnegative integer and x₁ + x₂ + x3 + x4 + 3x5 = 12. 2) We will buy 3 pies (not necessarily all different) from a store that sells 4 kinds of pie. different ord possible? List all of the possibilities (using A for apple

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.