A cylindrical piece of steel 57 mm (21 4 in.) in diameter is to be austenitized and quenched such that a minimum hardness of 45 HRC is to be produced throughout the entire piece. Of the alloys 8660, 8640, 8630, and 8620, which qualifies if the quenching medium is (a) moderately agitated water or (b) moderately agitated oil? Justify your choice(s).

Definition 1.1. Given two integers a and d with d non-zero, we say that d divides a (written d | a) if there is an integer c with a = cd. If no such integer exists, so d does not divide a, we write d - a. If d divides a, we say that d is a divisor of a. Proposition 1.2.1: Assume that a, b, and c are integers. If a | b and b | c, then a | c. Proposition 1.3. Assume that a, b, d, x, and y are integers. If d | a and d | b, then d | ax + by. Corollary 1.4. Assume that a, b, and d are integers. If d | a and d | b, then d | a + b and d | a − b. Proposition 1.4. Let a, b, c ∈ Z be integers. a) If a | b and b | c, then a | c. b) If a | b and b | a, then a = ±b. c) If a | b and a | c, then a | (b + c) and a | (b − c). Prime: A prime number is an integer p ≥ 2 whose only divisors are