Prove or disprove each of the following statements. (a) there exists a prime number x such that x + 16 and x + 32 are also prime numbers. (b) va, b, c, m € z+, if a = b (mod m), then c = c (mod m). (c) for any positive odd integer n, 3]n or nº = 1 (mod 12). (d) there exist 100 consecutive composite integers.

Acaterer charges $500 plus $30 per guest to cater a wedding. walt and traci don't want to spend more than $8000 on catering. write and solve an inequality in terms of the number of guests, g, that can be invited. a) 30g ? 8000; g ? 267 b) 500g < 8000; g < 16 c) 500 + 30g < 8000; g < 250 d) 500 + 30g ? 8000; g ? 250