Prime Numbers, Division, And Profit Calculation A Mathematical Analysis

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This article delves into fundamental mathematical concepts, addressing questions related to prime numbers, division, and profit calculation. We will explore the properties of prime numbers, perform division operations, and analyze profit figures. This exploration aims to solidify understanding and enhance problem-solving skills in mathematics.

Q9) Identifying the Smallest Odd Prime Number

Prime numbers are the cornerstone of number theory, playing a vital role in cryptography, computer science, and various mathematical fields. Prime numbers, by definition, are whole numbers greater than 1 that have only two distinct divisors: 1 and themselves. This means a prime number cannot be evenly divided by any other number except 1 and itself. Understanding this definition is crucial for identifying prime numbers and distinguishing them from composite numbers, which have more than two divisors.

To identify the smallest odd prime number, we must first clarify what "odd" signifies. Odd numbers are integers that are not divisible by 2, leaving a remainder of 1 when divided by 2. This excludes even numbers like 2, 4, 6, and so on. Therefore, we need to find a prime number that also fits the criterion of being an odd number.

Now, let's evaluate the options provided:

  • (a) 6: The number 6 is an even number (divisible by 2) and is also divisible by 1, 2, 3, and 6. Thus, it is not a prime number.
  • (b) 3: The number 3 is an odd number, and its only divisors are 1 and 3. This aligns perfectly with the definition of a prime number. Therefore, 3 is a prime number.
  • (c) 1: The number 1 is a special case. By convention, 1 is neither considered a prime nor a composite number. It only has one divisor (itself), which doesn't meet the requirement of having exactly two distinct divisors for a prime number.
  • (d) 2: The number 2 is the smallest prime number, but it is also the only even prime number. While it fits the definition of a prime number (divisible only by 1 and 2), it's not an odd number.

Considering these evaluations, we can confidently conclude that the smallest odd prime number among the given options is 3. The number 3 fulfills both criteria: it's a prime number (only divisible by 1 and itself) and an odd number (not divisible by 2). This understanding underscores the importance of grasping the fundamental definitions and properties of numbers in mathematical problem-solving.

Q10) Performing Division: 2,070 Divided by 23

Division is one of the four basic arithmetic operations, and it involves splitting a quantity into equal parts or groups. In this question, we need to divide 2,070 by 23. This mathematical operation will help us determine how many times 23 fits into 2,070, which is also known as finding the quotient. Mastering division is fundamental to many mathematical concepts and real-world applications, including resource allocation, measurement, and data analysis.

To perform the division, we can use long division or a calculator. Let's walk through the long division process for clarity:

  1. Set up the division problem: Write 2,070 (the dividend) inside the division symbol and 23 (the divisor) outside.
  2. Divide the first digits: 23 does not go into 2, so we consider the first two digits, 20. 23 still doesn't go into 20, so we move to the first three digits, 207.
  3. Estimate the quotient: Estimate how many times 23 goes into 207. A reasonable estimate is 9 times (since 23 * 9 is close to 207).
  4. Multiply and subtract: Multiply 23 by 9, which equals 207. Subtract 207 from 207, resulting in 0.
  5. Bring down the next digit: Bring down the next digit from the dividend, which is 0. We now have 0 as the remainder.
  6. Divide the remaining number: Divide 0 by 23, which equals 0.

Alternatively, using a calculator, we can directly compute 2,070 ÷ 23. The result is 90. This confirms our long division calculation and highlights the efficiency of using calculators for complex arithmetic operations.

Therefore, when 2,070 is divided by 23, the quotient is 90. This quotient represents the number of times 23 can be completely contained within 2,070. Understanding division and its applications is crucial not only in academic settings but also in everyday situations where fair distribution, proportional scaling, and quantitative analysis are required.

Q11) Calculating Total Profit Over Three Years

Calculating profit is a crucial aspect of financial analysis, especially in business and economics. Profit represents the financial gain realized when revenue exceeds expenses. In this question, we are tasked with finding the total profit a company made over three years. To do this, we must sum the profits earned in each individual year. This process of calculating total profit is a fundamental practice for assessing a company's financial performance and stability. Understanding how to calculate profit is essential for business owners, investors, and anyone interested in financial literacy.

The question provides the profit earned by a company in three consecutive years:

  • First year: ₹ 7,89,325
  • Second year: ₹ 6,12,478
  • Third year: ₹ 5,45,739

To find the total profit, we simply add these amounts together:

Total Profit = Profit in First Year + Profit in Second Year + Profit in Third Year

Total Profit = ₹ 7,89,325 + ₹ 6,12,478 + ₹ 5,45,739

To perform the addition, we align the numbers vertically by place value (ones, tens, hundreds, etc.) and add each column, carrying over to the next column when necessary. Let's break down the addition step by step:

  • Add the ones column: 5 + 8 + 9 = 22. Write down 2 and carry over 2 to the tens column.
  • Add the tens column: 2 (carried over) + 2 + 7 + 3 = 14. Write down 4 and carry over 1 to the hundreds column.
  • Add the hundreds column: 1 (carried over) + 3 + 4 + 7 = 15. Write down 5 and carry over 1 to the thousands column.
  • Add the thousands column: 1 (carried over) + 9 + 2 + 5 = 17. Write down 7 and carry over 1 to the ten-thousands column.
  • Add the ten-thousands column: 1 (carried over) + 8 + 1 + 4 = 14. Write down 4 and carry over 1 to the lakhs column.
  • Add the lakhs column: 1 (carried over) + 7 + 6 + 5 = 19. Write down 9 and carry over 1 to the ten-lakhs column.
  • Since there are no more columns, write down the carried-over 1.

Adding these amounts together, we get:

₹ 7,89,325 ₹ 6,12,478 ₹ 5,45,739

₹ 19,47,542

Therefore, the total profit the company made over the three years is ₹ 19,47,542. This calculation provides a clear picture of the company's cumulative financial performance over the given period. Understanding how to add and analyze financial figures like profit is a valuable skill for anyone involved in business or finance.

In conclusion, this article explored essential mathematical concepts through a series of questions. We identified the smallest odd prime number, performed division to find a quotient, and calculated total profit over three years. These exercises underscore the importance of mastering fundamental mathematical operations and their applications in real-world scenarios.