<p>If the fractional part of the number \(\dfrac{2^{403}}{15}\) is \(\dfrac{k}{15}\), then \(k\) is equal to ___________.</p>
Step-by-Step Solution
Key Concept: Use the Binomial Theorem to expand 2^403 = (15-13)^403 and find the remainder when divided by 15. The fractional part's numerator k is this remainder.
<p><strong>Step 1:</strong> Express 2^403 using Binomial Theorem. Write 2^403 = (15-13)^403</p><p><strong>Step 2:</strong> Expand using Binomial Theorem: (15-13)^403 = Σ C(403,r)·15^r·(-13)^(403-r)</p><p><strong>Step 3:</strong> When divided by 15, all terms with r ≥ 1 contain factor 15 and vanish. Only the r=0 term remains: (-13)^403</p><p><strong>Step 4:</strong> Calculate (-13)^403 ≡ -13^403 (mod 15). Since 13 ≡ -2 (mod 15), we have: (-13)^403 ≡ -(-2)^403 ≡ -(-2^403) ≡ 2^403 (mod 15)</p><p><strong>Step 5:</strong> Find 2^403 (mod 15). Note: 2^4 = 16 ≡ 1 (mod 15). So 403 = 4(100) + 3, thus 2^403 ≡ 2^3 ≡ 8 (mod 15)</p><p><strong>Step 6:</strong> Therefore 2^403 = 15q + 8 for some integer q, giving fractional part = 8/15</p><p>∴ k = <strong>8</strong></p>
Correct Answer: 8