Basic Mathematics & Logarithm
Number Theory
Grade 11
Question:
<p>Suppose, the seed of any positive integer \(n\) is defined as follows: Seed\((n) = n\), if \(n < 10\) = seed\((s(n))\), otherwise, where \(s(n)\) indicates the sum of digits of \(n\). For example, seed\((7) = 7\), seed\((248)\) = seed\((2+4+8)\) = seed\((14)\) = seed\((1+4)\) = seed\((5) = 5\) etc. How many positive integers \(n\), such that \(n < 500\), will have seed\((n) = 9\)?</p>
<p>(a) 39</p>
<p>(b) 72</p>
<p>(c) 81</p>
<p>(d) 55</p>
Step-by-Step Solution
Key Concept: The seed function recursively sums digits until reaching a single digit. Recognize that seed(n) equals n mod 9 (with seed(n)=9 when n≡0 mod 9), which is why we can find patterns without computing the entire sequence.
<p><strong>Key Insight:</strong> The seed function is equivalent to finding the digital root of n. Mathematically, seed(n) = 1 + ((n-1) mod 9) for n ≥ 1, or equivalently: seed(n) ≡ n (mod 9), where seed(n) ∈ {1,2,...,9}.</p><p><strong>Step 1:</strong> For any positive integer n, seed(n) = n mod 9 if n ≢ 0 (mod 9), and seed(n) = 9 if n ≡ 0 (mod 9).</p><p><strong>Step 2:</strong> This property allows us to find seed values without iterative digit summation. For example: seed(47) = 4+7=11 → 1+1=2, or directly: 47 mod 9 = 2.</p><p><strong>Step 3:</strong> To find which numbers have a specific seed value k, we need n ≡ k (mod 9). For seed(n) = 5: n ∈ {5, 14, 23, 32, 41, 50, ...}.</p><p>∴ Answer: D</p>
Correct Answer: D