Proofs in Mathematics

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Step 1: Let the two even numbers be 2m and 2n (where m, n are integers). Step 2: Sum = 2m + 2n Step 3: Factor out 2 → Sum = 2(m + n) Step 4: Since (m + n) is an integer, let k = m + n Step 5: Sum = 2k

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Step 1: Look for cases where product might be smaller Step 2: Try negative integers: Take -2 and 3 Step 3: Product = (-2) × 3 = -6 Step 4: Check: -6 < -2 and -6 < 3 Step 5: Since -6 is less than both

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Step 1: Let n be any odd integer Step 2: Write n = 2k + 1 (where k is an integer) Step 3: Calculate n² = (2k + 1)² Step 4: Expand: n² = 4k² + 4k + 1 Step 5: Factor: n² = 4k(k + 1) + 1 = 2[2k(k + 1)] +

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Step 1: Identify the given statements - Statement 1: All prime numbers > 2 are odd - Statement 2: 17 is a prime number > 2 Step 2: Apply deductive reasoning Step 3: Since 17 fits the condition (prime

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Step 1: Draw triangle ABC Step 2: Extend side BC to point D Step 3: Draw line CE parallel to AB through C Step 4: ∠BAC = ∠ACE (alternate angles, AB || CE) Step 5: ∠ABC = ∠ECD (corresponding angles, AB

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Step 1: Identify the reasoning type - This is inductive reasoning Step 2: Analyze the sample size - Only 3 cats observed Step 3: Check generalization - Claims about ALL cats from limited sample Step 4

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Step 1: Let n be any integer Step 2: Calculate the product: 2n Step 3: By definition of even numbers: An even number is of the form 2k where k is an integer Step 4: In our case, 2n is exactly in the f

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Step 1: Observe the pattern: 1² = 1 11² = 121 111² = 12321 Step 2: Notice pattern in results: - Digits go 1,2,1 then 1,2,3,2,1 Step 3: Conjecture: 1111² = 1234321 Step 4: Verify by calculation: 1111²