<p>Find the total number of positive integral solutions for \((x, y, z)\) such that \(xyz = 24\). Also find the total number of integral solutions.</p>
Step-by-Step Solution
Key Concept: For positive integral solutions, find all ordered factorizations of 24 into three factors by using its prime factorization (2³ × 3). For all integral solutions, account for sign combinations: each positive solution generates 2³ = 8 sign variants (each variable can be ±).
<p><strong>Step 1: Prime Factorization</strong></p><p>24 = 2³ × 3¹</p><p><strong>Step 2: Positive Integral Solutions</strong></p><p>We need xyz = 24 where x, y, z are positive integers. We distribute the prime factors among x, y, z:</p><p>• For factor 2³: Distribute 3 powers of 2 among x, y, z in (a,b,c) form where a+b+c=3, a,b,c ≥ 0. Number of ways = C(3+3-1,3-1) = C(5,2) = 10</p><p>• For factor 3¹: Distribute 1 power of 3 among x, y, z in (p,q,r) form where p+q+r=1, p,q,r ≥ 0. Number of ways = C(1+3-1,3-1) = C(3,2) = 3</p><p>Total positive solutions = 10 × 3 = <strong>30</strong></p><p><strong>Step 3: Integral Solutions</strong></p><p>For each positive solution (x, y, z), we can independently choose the sign of each variable:</p><p>• x can be ±x (2 choices)</p><p>• y can be ±y (2 choices)</p><p>• z can be ±z (2 choices)</p><p>But xyz = 24 > 0, so the product of signs must be positive. This means an even number of negative variables (0 or 2 negatives).</p><p>Sign combinations: (+,+,+), (−,−,+), (−,+,−), (+,−,−) = 4 ways per positive solution</p><p>Total integral solutions = 30 × 4 = <strong>120</strong></p>
Correct Answer: Positive integral solutions: 30; Integral solutions: 120