This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the questions: Step 1: Answer question 1. To find the number of integers from 27000 to 106780 (inclusive), we use the formula: Last number - First number + 1. Number of integers = 106780 - 27000 + 1 Number of integers = 79780 + 1 Number of integers = 79781 The number of integers is 79781. Step 2: Answer question 2. To find the sum of these numbers, we use the formula for the sum of an arithmetic progression: S_n = (n)/(2)(a + l), where n is the number of terms, a is the first term, and l is the last term. Here, n = 79781, a = 27000, and l = 106780. S_n = (79781)/(2)(27000 + 106780) S_n = (79781)/(2)(133780) S_n = 79781 × 66890 S_n = 5336180090 The sum of the numbers is 5,336,180,090. Step 3: Answer question 3. Mark the statements as correct (✓) or incorrect (X): (i) The smallest prime number is 1. ❌ (The smallest prime number is 2.) (ii) The smallest even prime number is 2. ✓ (iii) A factor of a number cannot be greater than that number. ✓ (iv) The H.C.F. of two prime numbers is 1. ✓ (v) The H.C.F. of two even numbers will always be an even number. ✓ Step 4: Answer question 4. The question asks for the square root (ବର୍ଗମୂଳ) of the given numbers. However, some numbers are perfect cubes rather than perfect squares. Assuming the question intends for integer roots (square or cube) where they exist: For 1331: This is 11^3. The cube root is [3]1331 = 11. For 6084: This is 78^2. The square root is sqrt(6084) = 78. For 4913: This is 17^3. The cube root is [3]4913 = 17. For 12167: This is 23^3. The cube root is [3]12167 = 23. For 32768: This is 32^3. The cube root is [3]32768 = 32. For 8000: This is 20^3. The cube root is [3]8000 = 20. That's 2 down. 3 left today — send the next one.