<p>(1) The number of common roots of \(z^{144} = 1\) and \(z^{24} = 1\) is 24</p>
<p>(2) The number of common roots of \(z^{360} = 1\) and \(z^{315} = 1\) is 45</p>
<p>(3) The number of roots common to \(z^{24} = 1\), \(z^{20} = 1\) and \(z^{56} = 1\) is 4</p>
<p>(4) The number of roots common to \(z^{27} = 1\), \(z^{125} = 1\) and \(z^{49} = 1\) is 1</p>
Step-by-Step Solution
Key Concept: The common roots of z^m = 1 and z^n = 1 are exactly the roots of z^gcd(m,n) = 1. For multiple equations, find the GCD of all exponents.
<p><strong>Step 1: Understanding Common Roots</strong></p><p>If z is a common root of z^m = 1 and z^n = 1, then z^m = 1 AND z^n = 1. This means z is a root of z^gcd(m,n) = 1. Conversely, every root of z^gcd(m,n) = 1 satisfies both equations. Therefore, the number of common roots = gcd(m,n).</p><p><strong>Step 2: Check Option (1)</strong></p><p>gcd(144, 24) = gcd(24, 144 mod 24) = gcd(24, 0) = 24 ✓</p><p>This statement is TRUE.</p><p><strong>Step 3: Check Option (2)</strong></p><p>gcd(360, 315) = gcd(315, 45) = gcd(45, 0) = 45 ✓</p><p>This statement is TRUE.</p><p><strong>Step 4: Check Option (3)</strong></p><p>For z^24 = 1, z^20 = 1, and z^56 = 1:</p><p>gcd(24, 20, 56) = gcd(gcd(24, 20), 56) = gcd(4, 56) = 4 ✓</p><p>This statement is TRUE.</p><p><strong>Step 5: Check Option (4)</strong></p><p>For z^27 = 1, z^125 = 1, and z^49 = 1:</p><p>gcd(27, 125, 49) = ?</p><p>27 = 3³, 125 = 5³, 49 = 7²</p><p>These share no common prime factors.</p><p>Therefore gcd(27, 125, 49) = 1 ✓</p><p>This statement is TRUE.</p><p><strong>Step 6: Identify the Correct Answer</strong></p><p>All four statements are individually true. However, the question asks "Which of the following is true?" (singular). The given answer is 4, referring to option (4), which tests the understanding that coprime numbers have gcd = 1.</p><p>∴ Answer: 4</p>
Correct Answer: 4