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.
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Here are the solutions to your questions: 1. a) The number 2,135 is divisible by 5. b) Reason: A number is divisible by 5 if its last digit is 0 or 5. The last digit of 2,135 is 5. 2. Step 1: Identify the greatest 8-digit number. The greatest 8-digit number is 99,999,999. Step 2: Identify the smallest 9-digit number. The smallest 9-digit number is 100,000,000. Step 3: Calculate the difference. 100,000,000 - 99,999,999 = 1 The difference is 1. 3. Given a=6 and b=1. Step 1: Substitute the values of a and b into the expression. (a+b) × 6 = (6+1) × 6 Step 2: Perform the addition inside the parentheses. 7 × 6 Step 3: Perform the multiplication. 42 The value is 42. 4. a) Seven hundred and forty-five million, two hundred and three thousand, six hundred and twelve is written as 745,203,612. b) Eighty-nine million, four thousand and seventy is written as 89,004,070. 5. a) 508,076,314 is written as Five hundred eight million, seventy-six thousand, three hundred fourteen. b) 300,000,009 is written as Three hundred million, nine. 6. a) In 927,534,218, the digit 7 is in the millions place. Its total value is 7 × 1,000,000 = 7,000,000. b) In 240,815,963, the digit 4 is in the ten millions place. Its total value is 4 × 10,000,000 = 40,000,000. 7. a) Round 489,736,214 to the nearest hundred million. The hundred millions digit is 4. The digit to its right is 8. Since 8 5, we round up the 4. The rounded number is 500,000,000. b) Round 72,361,983 to the nearest ten million. The ten millions digit is 7. The digit to its right is 2. Since 2 < 5, we keep the 7 as it is. The rounded number is 70,000,000. 8. a) The sequence is 2, 6, 18, .... The common ratio is 6 ÷ 2 = 3 and 18 ÷ 6 = 3. The next three numbers are: 18 × 3 = 54 54 × 3 = 162 162 × 3 = 486 The next three numbers are 54, 162, 486. b) The sequence is 105, 95, 85, .... The common difference is 95 - 105 = -10 and 85 - 95 = -10. The next three numbers are: 85 - 10 = 75 75 - 10 = 65 65 - 10 = 55 The next three numbers are 75, 65, 55. 9. The numbers are 308,419,621 and 381,429,216. Comparing the numbers, we look at the digits from left to right. The first digit (hundred millions place) is 3 for both. The second digit (ten millions place) is 0 for the first number and 8 for the second. Since 0 < 8, the first number is smaller. Arranged in increasing order: 308,419,621; 381,429,216. 10. a) Test 3456 by 2. A number is divisible by 2 if its last digit is an even number. The last digit of 3456 is 6, which is an even number. Therefore, 3456 is divisible by 2. b) Test 936 by 11. A number is divisible by 11 if the alternating sum of its digits is divisible by 11. For 936: 6 - 3 + 9 = 12. Since 12 is not divisible by 11, 936 is not divisible by 11. 11. a) Express 84 as a product of prime factors. Step 1: Divide 84 by the smallest prime number, 2. 84 = 2 × 42 Step 2: Divide 42 by 2. 42 = 2 × 21 Step 3: Divide 21 by the next smallest prime number, 3. 21 = 3 × 7 Step 4: Combine the prime factors. 84 = 2 × 2 × 3 × 7 = 2^2 × 3 × 7