Calculate the positive square root of 100 using the prime factor method, and arrange the following numbers in descending order: 69831, 69813, 98813, 96831, 86931.

Question

Asked on April 1, 2026Mathematics

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.

Accepted Answer · 2026-04-01T12:51:23.375Z

Here are the solutions to your problems: 1. Positive square root using prime factor method Step 1: Identify all prime numbers greater than 1 and less than 25. The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23. Step 2: Calculate the sum of these prime numbers. 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100 Step 3: Find the prime factorization of the sum (100). 100 = 2 × 50 100 = 2 × 2 × 25 100 = 2 × 2 × 5 × 5 100 = 2^2 × 5^2 Step 4: Obtain the positive square root using the prime factors. sqrt(100) = sqrt(2^2 × 5^2) sqrt(100) = 2 × 5 sqrt(100) = 10 The positive square root is 10. 2. Arrange numbers in descending order Step 1: List the given numbers: 69831, 69813, 98813, 96831, 86931. Step 2: Compare the numbers starting from the leftmost digit (tens of thousands place). The largest digit in the tens of thousands place is 9. The numbers starting with 9 are 98813 and 96831. Comparing 98813 and 96831: 98813 > 96831. The next largest digit in the tens of thousands place is 8. The number is 86931. The next largest digit in the tens of thousands place is 6. The numbers starting with 6 are 69831 and 69813. Comparing 69831 and 69813: 69831 > 69813. Step 3: Arrange them in descending order (largest to smallest). 98813, 96831, 86931, 69831, 69813 The numbers in descending order are 98813, 96831, 86931, 69831, 69813. 3. Evaluate fractions Step 1: Find the least common multiple (LCM) of the denominators 2, 3, 4, 5. The LCM of 2, 3, 4, 5 is 60. Step 2: Convert each fraction to an equivalent fraction with a denominator of 60. (1)/(2) = (1 × 30)/(2 × 30) = (30)/(60) (1)/(3) = (1 × 20)/(3 × 20) = (20)/(60) (1)/(4) = (1 × 15)/(4 × 15) = (15)/(60) (1)/(5) = (1 × 12)/(5 × 12) = (12)/(60) Step 3: Perform the operations. (30)/(60) - (20)/(60) + (15)/(60) - (12)/(60) = (30 - 20 + 15 - 12)/(60) = (10 + 15 - 12)/(60) = (25 - 12)/(60) = (13)/(60) The answer in improper fraction form is (13)/(60). 4. Value of expression and place value Step 1: Perform the subtraction 4986093 - 948699. 4986093 948699 ---------- 4037394 Step 2: Perform the division 4037394 ÷ 125. Using long division: 4037394 ÷ 125 = 32299.152 Since the question implies an integer division context (and place value of digit 1), let's re-check the problem statement. "Find without using a calculator, the value of (4986093-948699) ÷ 125". This implies an exact division or a result where the place value of 1 is clear. Let's assume the question expects an integer result, which means there might be a typo in the numbers or the division is not exact. However, if we perform the division as given: 4037394 ÷ 125 125 × 3 = 375 403 - 375 = 28 287 ÷ 125 = 2 remainder 37 373 ÷ 125 = 2 remainder 123 1239 ÷ 125 = 9 remainder 114 1144 ÷ 125 = 9 remainder 19 So, 4037394 ÷ 125 = 32299 with a remainder of 19. If the question expects an exact value, it would be 32299 (19)/(125) or 32299.152. Given the instruction "without using a calculator" and "place value of digit 1", it's likely an integer answer is expected, or the question implies a specific context for the result. Let's assume the result is 32299 (integer part). If we consider the exact decimal value: 32299.152. The digit '1' is in the tenths place. If we consider the integer part 32299, there is no digit '1'. This suggests there might be an error in my interpretation or the problem itself. Let's re-evaluate the subtraction and division carefully. 4986093 - 948699 = 4037394. 4037394 ÷ 125. Let's perform long division: ` 32299 _______ 125 | 4037394 -375 ---- 287 -250 ---- 373 -250 ---- 1239 -1125 ----- 1144 -1125 ----- 19 ` So, 4037394 ÷ 125 = 32299 with a remainder of 19. The question asks for "the value of" and then "place value of digit 1 in the answer obtained above". This implies the answer should contain a '1'. Let's consider if the question implies rounding or if there's a specific context for the answer. If the answer is 32299.152, then the digit '1' is in the tenths place. If the question implies an integer answer, then there is no '1'. Given the context of "without using a calculator", it's common for such problems to have exact integer answers. Let's re-check the numbers for any common factors or if the numbers are designed to give a specific result. 125 = 5^3. For a number to be exactly divisible by 125, its last three digits must be divisible by 125. 394 is not divisible by 125. So the division is not exact. If the question expects an integer answer, and there is no '1' in 32299, then the question about the place value of '1' is problematic. However, if we consider the decimal representation 32299.152: The digit '1' is in the first decimal place, which is the tenths place. Let's assume the question expects the decimal value and then asks for the place value of '1'. The value is 32299.152. The digit '1' in 32299.152 is in the tenths place. 5. Consecutive odd integers a) Write the four integers in ascending order in terms of x. Step 1: Define the four consecutive odd integers. If the smallest odd integer is x, the next consecutive odd integers are x+2, x+4, x+6. The four integers in ascending order are x, x+2, x+4, x+6. b) Determine the value of x+6. Step 1: Set up an equation using the sum of the four integers. The sum of the four consecutive odd integers is 392. x + (x+2) + (x+4) + (x+6) = 392 Step 2: Solve for x. 4x + 12 = 392 4x = 392 - 12 4x = 380 x = (380)/(4) x = 95 Step 3: Determine the value of x+6. x+6 = 95+6 x+6 = 101 The value of x+6 is 101. 6. Product of numbers divisible by 4 and 6 Step 1: Understand the divisibility rule. A number is divisible by both 4 and 6 if it is divisible by their least common multiple, which is 12. Divisibility by 12 means: • Divisible by 4 (last two digits form a number divisible by 4). • Divisible by 3 (sum of digits is divisible by 3). Step 2: Check each number: 126, 162, 612, 462, 336, 632. • 126: Divisible by 4? No, 26 is not divisible by 4. (Reject) • 162: Divisible by 4? No, 62 is not divisible by 4. (Reject) • 612: Divisible by 4? Yes, 12 is divisible by 4. Divisible by 3? Yes, 6+1+2 = 9, which is divisible by 3. So, 612 is divisible by 12. (Keep) • 462: Divisible by 4? No, 62 is not divisible by 4. (Reject) • 336: Divisible by 4? Yes, 36 is divisible by 4. Divisible by 3? Yes, 3+3+6 = 12, which is divisible by 3. So, 336 is divisible by 12. (Keep) • 632: Divisible by 4? Yes, 32 is divisible by 4. Divisible by 3? No, 6+3+2 = 11, which is not divisible by 3. (Reject) The numbers divisible by both 4 and 6 are 612 and 336. Step 3: Find the product of these numbers. 612 × 336 612 × 336 = 205632 The product is 205632. 7. Prime factorization and quotient Step 1: Express 4719 as the product of its prime factors. Start dividing by prime numbers: 4719 ÷ 3 = 1573 Now, check for prime factors of 1573. 1573 ÷ 11 = 143 143 ÷ 11 = 13 So, 4719 = 3 × 11 × 11 × 13 = 3 × 11^2 × 13. Step 2: Express 7623 as the product of its prime factors. Start dividing

Accepted Answer · 2026-04-01T12:51:23.375Z

Here are the solutions to your problems: 1. Positive square root using prime factor method Step 1: Identify all prime numbers greater than 1 and less than 25. The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23. Step 2: Calculate the sum of these prime numbers. 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100 Step 3: Find the prime factorization of the sum (100). 100 = 2 × 50 100 = 2 × 2 × 25 100 = 2 × 2 × 5 × 5 100 = 2^2 × 5^2 Step 4: Obtain the positive square root using the prime factors. sqrt(100) = sqrt(2^2 × 5^2) sqrt(100) = 2 × 5 sqrt(100) = 10 The positive square root is 10. 2. Arrange numbers in descending order Step 1: List the given numbers: 69831, 69813, 98813, 96831, 86931. Step 2: Compare the numbers starting from the leftmost digit (tens of thousands place). The largest digit in the tens of thousands place is 9. The numbers starting with 9 are 98813 and 96831. Comparing 98813 and 96831: 98813 > 96831. The next largest digit in the tens of thousands place is 8. The number is 86931. The next largest digit in the tens of thousands place is 6. The numbers starting with 6 are 69831 and 69813. Comparing 69831 and 69813: 69831 > 69813. Step 3: Arrange them in descending order (largest to smallest). 98813, 96831, 86931, 69831, 69813 The numbers in descending order are 98813, 96831, 86931, 69831, 69813. 3. Evaluate fractions Step 1: Find the least common multiple (LCM) of the denominators 2, 3, 4, 5. The LCM of 2, 3, 4, 5 is 60. Step 2: Convert each fraction to an equivalent fraction with a denominator of 60. (1)/(2) = (1 × 30)/(2 × 30) = (30)/(60) (1)/(3) = (1 × 20)/(3 × 20) = (20)/(60) (1)/(4) = (1 × 15)/(4 × 15) = (15)/(60) (1)/(5) = (1 × 12)/(5 × 12) = (12)/(60) Step 3: Perform the operations. (30)/(60) - (20)/(60) + (15)/(60) - (12)/(60) = (30 - 20 + 15 - 12)/(60) = (10 + 15 - 12)/(60) = (25 - 12)/(60) = (13)/(60) The answer in improper fraction form is (13)/(60). 4. Value of expression and place value Step 1: Perform the subtraction 4986093 - 948699. 4986093 948699 ---------- 4037394 Step 2: Perform the division 4037394 ÷ 125. Using long division: 4037394 ÷ 125 = 32299.152 Since the question implies an integer division context (and place value of digit 1), let's re-check the problem statement. "Find without using a calculator, the value of (4986093-948699) ÷ 125". This implies an exact division or a result where the place value of 1 is clear. Let's assume the question expects an integer result, which means there might be a typo in the numbers or the division is not exact. However, if we perform the division as given: 4037394 ÷ 125 125 × 3 = 375 403 - 375 = 28 287 ÷ 125 = 2 remainder 37 373 ÷ 125 = 2 remainder 123 1239 ÷ 125 = 9 remainder 114 1144 ÷ 125 = 9 remainder 19 So, 4037394 ÷ 125 = 32299 with a remainder of 19. If the question expects an exact value, it would be 32299 (19)/(125) or 32299.152. Given the instruction "without using a calculator" and "place value of digit 1", it's likely an integer answer is expected, or the question implies a specific context for the result. Let's assume the result is 32299 (integer part). If we consider the exact decimal value: 32299.152. The digit '1' is in the tenths place. If we consider the integer part 32299, there is no digit '1'. This suggests there might be an error in my interpretation or the problem itself. Let's re-evaluate the subtraction and division carefully. 4986093 - 948699 = 4037394. 4037394 ÷ 125. Let's perform long division: ` 32299 _______ 125 | 4037394 -375 ---- 287 -250 ---- 373 -250 ---- 1239 -1125 ----- 1144 -1125 ----- 19 ` So, 4037394 ÷ 125 = 32299 with a remainder of 19. The question asks for "the value of" and then "place value of digit 1 in the answer obtained above". This implies the answer should contain a '1'. Let's consider if the question implies rounding or if there's a specific context for the answer. If the answer is 32299.152, then the digit '1' is in the tenths place. If the question implies an integer answer, then there is no '1'. Given the context of "without using a calculator", it's common for such problems to have exact integer answers. Let's re-check the numbers for any common factors or if the numbers are designed to give a specific result. 125 = 5^3. For a number to be exactly divisible by 125, its last three digits must be divisible by 125. 394 is not divisible by 125. So the division is not exact. If the question expects an integer answer, and there is no '1' in 32299, then the question about the place value of '1' is problematic. However, if we consider the decimal representation 32299.152: The digit '1' is in the first decimal place, which is the tenths place. Let's assume the question expects the decimal value and then asks for the place value of '1'. The value is 32299.152. The digit '1' in 32299.152 is in the tenths place. 5. Consecutive odd integers a) Write the four integers in ascending order in terms of x. Step 1: Define the four consecutive odd integers. If the smallest odd integer is x, the next consecutive odd integers are x+2, x+4, x+6. The four integers in ascending order are x, x+2, x+4, x+6. b) Determine the value of x+6. Step 1: Set up an equation using the sum of the four integers. The sum of the four consecutive odd integers is 392. x + (x+2) + (x+4) + (x+6) = 392 Step 2: Solve for x. 4x + 12 = 392 4x = 392 - 12 4x = 380 x = (380)/(4) x = 95 Step 3: Determine the value of x+6. x+6 = 95+6 x+6 = 101 The value of x+6 is 101. 6. Product of numbers divisible by 4 and 6 Step 1: Understand the divisibility rule. A number is divisible by both 4 and 6 if it is divisible by their least common multiple, which is 12. Divisibility by 12 means: • Divisible by 4 (last two digits form a number divisible by 4). • Divisible by 3 (sum of digits is divisible by 3). Step 2: Check each number: 126, 162, 612, 462, 336, 632. • 126: Divisible by 4? No, 26 is not divisible by 4. (Reject) • 162: Divisible by 4? No, 62 is not divisible by 4. (Reject) • 612: Divisible by 4? Yes, 12 is divisible by 4. Divisible by 3? Yes, 6+1+2 = 9, which is divisible by 3. So, 612 is divisible by 12. (Keep) • 462: Divisible by 4? No, 62 is not divisible by 4. (Reject) • 336: Divisible by 4? Yes, 36 is divisible by 4. Divisible by 3? Yes, 3+3+6 = 12, which is divisible by 3. So, 336 is divisible by 12. (Keep) • 632: Divisible by 4? Yes, 32 is divisible by 4. Divisible by 3? No, 6+3+2 = 11, which is not divisible by 3. (Reject) The numbers divisible by both 4 and 6 are 612 and 336. Step 3: Find the product of these numbers. 612 × 336 612 × 336 = 205632 The product is 205632. 7. Prime factorization and quotient Step 1: Express 4719 as the product of its prime factors. Start dividing by prime numbers: 4719 ÷ 3 = 1573 Now, check for prime factors of 1573. 1573 ÷ 11 = 143 143 ÷ 11 = 13 So, 4719 = 3 × 11 × 11 × 13 = 3 × 11^2 × 13. Step 2: Express 7623 as the product of its prime factors. Start dividing

Calculate the positive square root of 100 using the prime factor method, and arrange the following numbers in descending order: 69831, 69813, 98813, 96831, 86931.

Need help with your own homework?

More Mathematics Questions

Calculate the positive square root of 100 using the prime factor method, and arrange the following numbers in descending order: 69831, 69813, 98813, 96831, 86931.

Need help with your own homework?

More Mathematics Questions