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
Step 1: Sort the admission numbers as prime or composite. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A composite number is a natural number greater than 1 that is not prime. • For 201: 201 = 3 × 67. It has factors other than 1 and itself, so it is composite. • For 211: 211 is only divisible by 1 and 211. So, it is prime. • For 55: 55 = 5 × 11. It has factors other than 1 and itself, so it is composite. • For 112: 112 = 2 × 56. It has factors other than 1 and itself, so it is composite. • For 73: 73 is only divisible by 1 and 73. So, it is prime. Prime numbers: 211, 73 Composite numbers: 201, 55, 112 Step 2: Determine the number Onyango wrote down. a) The odd numbers from 27 to 32 are 27, 29, and 31. The product of these numbers is: 27 × 29 × 31 783 × 31 24273 The number Onyango wrote down is 24273. b) Determine the difference between the number of even digits and odd digits in 24273. The digits in 24273 are 2, 4, 2, 7, 3. Even digits: 2, 4, 2 (There are 3 even digits). Odd digits: 7, 3 (There are 2 odd digits). Difference = (Number of even digits) - (Number of odd digits) 3 - 2 = 1 The difference is 1. Step 3: Determine and write down the irrational numbers. An irrational number cannot be expressed as a simple fraction (p)/(q) and has a non-terminating, non-repeating decimal expansion. Square roots of non-perfect squares are irrational. • sqrt(79): Since 79 is not a perfect square (8^2=64, 9^2=81), sqrt(79) is an irrational number. • sqrt(2.197): This can be written as sqrt((2197)/(1000)). Since 2197 is not a perfect square (it ends in 7), sqrt(2197) is irrational. Therefore, sqrt(2.197) is an irrational number. • sqrt(30.42): This can be written as sqrt((3042)/(100)). Since 3042 is not a perfect square (it ends in 2), sqrt(3042) is irrational. Therefore, sqrt(30.42) is an irrational number. The irrational numbers from the list are sqrt(79), sqrt(2.197), sqrt(30.42). Step 4: Factorise 81 - (q-7)^2. This expression is in the form of a difference of squares, a^2 - b^2 = (a-b)(a+b). Here, a^2 = 81 a = 9. And b^2 = (q-7)^2 b = (q-7). Substitute these into the difference of squares formula: 81 - (q-7)^2 = [9 - (q-7)][9 + (q-7)] = (9 - q + 7)(9 + q - 7) = (16 - q)(2 + q) The factored form is (16 - q)(2 + q). Step 5: Express 729 and 2187 in index form, then evaluate (729)/(2187). First, find the prime factorization of 729: 729 ÷ 3 = 243 243 ÷ 3 = 81 81 ÷ 3 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 So, 729 = 3 × 3 × 3 × 3 × 3 × 3 = 3^6. Next, find the prime factorization of 2187: 2187 ÷ 3 = 729 Since 729 = 3^6, we have: 2187 = 3 × 3^6 = 3^7 Now, evaluate (729)/(2187): (729)/(2187) = (3^6)/(3^7) Using the exponent rule (a^m)/(a^n) = a^m-n: (3^6)/(3^7) = 3^6-7 = 3^-1 = (1)/(3) The index forms are 729 = 3^6 and 2187 = 3^7. The evaluation is (729)/(2187) = (1)/(3). That's 2 down. 3 left today — send the next one.