Model: NikolayKozloff/NuminaMath-7B-TIR-Q8_0-GGUF
Author: NikolayKozloff
Pipeline: text-generation
Tags: gguf, alignment-handbook, generated_from_trainer, llama-cpp, gguf-my-repo, text-generation, base_model:AI-MO/NuminaMath-7B-TIR, base_model:quantized:AI-MO/NuminaMath-7B-TIR, license:apache-2.0, endpoints_compatible, region:us, conversational
Base models (quantized): id: AI-MO/NuminaMath-7B-TIR
Card metadata: base_model: AI-MO/NuminaMath-7B-TIR
license: apache-2.0
pipeline_tag: text-generation
tags: alignment-handbook, generated_from_trainer, llama-cpp, gguf-my-repo
widget: example_title: Math problem
messages: user: For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
output: text: ### Solution: 1. For the polynomial \\( x^2 + kx + 36 \\) to have two distinct integer roots, let's denote these roots by \\( r_1 \\) and \\( r_2 \\).\n\n2. According to Vieta's formulas, the sum of the roots \\( r_1 + r_2 \\) is equal to \\(-k\\), and the product of the roots \\( r_1 \\cdot r_2 \\) is equal to 36.\n\n3. To find the distinct integer pairs \\((r_1, r_2)\\) whose product is 36, we need to determine all pairs of integers \\((r_1, r_2)\\) such that \\( r_1 \\cdot r_2 = 36 \\) and \\( r_1 \\neq r_2 \\).\n\n4. From the equation \\( r_1 + r_2 = -k \\), for each valid pair \\((r_1, r_2)\\), we can compute the corresponding value of \\( k \\).\n\n5. Finally, since we need the polynomial to have two distinct integer roots, we need to ensure that \\( r_1 \\) and \\( r_2 \\) are distinct.\n\nLet's start by finding all pairs \\((r_1, r_2)\\) such that \\( r_1 \\cdot r_2 = 36 \\). We'll then determine the values of \\( k \\) and ensure the roots are distinct.\n```python\nimport itertools\n\n# Find all pairs (r1, r2) such that r1 * r2 = 36\nproduct_36 = 36\nfactor_pairs = []\n\nfor i in range(1, product_36 + 1):\n if product_36 % i == 0:\n pair = (i, product_36 // i)\n if pair[0] != pair[1]: # Ensure distinct pairs\n factor_pairs.append(pair)\n\n# Calculate k for each pair and ensure distinct integer roots\nvalid_k_values = set()\nfor r1, r2 in factor_pairs:\n if r1 != r2:\n k = -(r1 + r2)\n valid_k_values.add(k)\n\nprint((len(valid_k_values), sorted(valid_k_values)))\n```\n```output\n(4, [-37, -20, -15, -13])\n```\nThe distinct integer values of \\( k \\) that make the polynomial \\( x^2 + kx + 36 \\) have two distinct integer roots are \\(-37, -20, -15, \\text{and} -13\\).\n\nTherefore, the number of such values of \\( k \\) is:\n\n\\[\n\\boxed{4}\n\\]\n
model-index: name: NuminaMath-7B-TIR
Likes: 1 | All-time downloads: 142
Created: 2024-07-10 21:19:27