Question

If \(\mathrm{X}=28+(1 \times 2 \times 3 \times 4 \times \ldots . \times 16 \times 28)\) and \(\mathrm{Y}=17+(1 \times 2 \times 3 \times \ldots \ldots \times 17)\), then which of the following is/are true? (a) \(\mathrm{X}\) is a composite number. (b) \(\mathrm{Y}\) is a prime number. (c) \(\mathrm{X}-\mathrm{Y}\) is a prime number. (d) \(\mathrm{X}+\mathrm{Y}\) is a composite number. (1) Both (a) and (d) (2) Both (b) and (c) (3) Both (b) and (d) (4) Both (a) and (b)

Short answer

(1) Both (a) and (d) (2) Both (b) and (c) (3) Both (b) and (d) (4) Both (a) and (b) Answer: (1) Both (a) and (d) are true.

Step-by-step solution

Find X and Y

First, we need to find the values of X and Y using the given expressions. The expressions can be simplified by factoring the common term (1 x 2 x 3 ...). For X: X = 28 + (1 × 2 × 3 × 4 × ... × 16 × 28) X = 28 + 28(1 × 2 × 3 × 4 × ... × 16) X = 28(1 + 1 × 2 × 3 × 4 × ... × 16) For Y: Y = 17 + (1 × 2 × 3 × ... × 17) Y = 17(1 + 1 × 2 × 3 × ... × 16)

Analyze statements (a) and (b)

Now that we have X and Y, we can examine statements (a) and (b). (a) X is a composite number. A composite number has factors other than 1 and itself. Since X is the product of 28 and some integers greater than 1, X has at least three factors (1, itself, and 28) and is a composite number. So, statement (a) is true. (b) Y is a prime number. A prime number has exactly two factors, 1 and itself. Similarly, as Y is the product of 17 and some integers greater than 1, it has at least 3 factors (1, 17, and itself) and is not a prime number. So, statement (b) is false.

Analyze statements (c) and (d)

Now, we move on to statements (c) and (d). (c) X - Y is a prime number. From step 1, we found that: X = 28(1 + 1 × 2 × 3 × 4 × ... × 16) Y = 17(1 + 1 × 2 × 3 × ... × 16) X - Y = 28(1 × 2 × 3 × ... × 16) - 17(1 × 2 × 3 × ... × 16) X - Y = (28 - 17)(1 × 2 × 3 × ... × 16) X - Y = 11(1 × 2 × 3 × ... × 16) Since 11 is a factor of (X-Y), it has more than two factors, which means it's not a prime number. Statement (c) is false. (d) X + Y is a composite number. X + Y = 28(1 × 2 × 3 × ... × 16) + 17(1 × 2 × 3 × ... × 16) X + Y = (28 + 17)(1 × 2 × 3 × ... × 16) X + Y = 45(1 × 2 × 3 × ... × 16) Since 45 is a factor of (X+Y), (X+Y) has more than two factors and is a composite number. Statement (d) is true.

Identify the correct answer

Now that we have analyzed the four statements, let's see which of the given options is correct: (1) Both (a) and (d) - True (2) Both (b) and (c) - False (3) Both (b) and (d) - False (4) Both (a) and (b) - False The correct answer is (1). Both statements (a) and (d) are true.

Key concepts

Composite Number

Understanding the concept of a composite number is vital for various mathematical disciplines, including number theory and algebra. A composite number is defined as a positive integer that has at least one positive divisor other than one and itself.

For example, the number 6 is composite because its divisors are 1, 2, 3, and 6. In contrast to prime numbers, which only have two distinct positive divisors—1 and themselves—composite numbers have more than two. This distinction is crucial when solving math problems involving factorization or determining the divisibility of numbers.

In the given exercise, we identify that the number X is composite because it is the product of 28 and other integers in the factorial sequence. This indicates that X has multiple factors beyond 1 and itself, confirming its status as a composite number.

Prime Number

Prime numbers are the building blocks of arithmetic, much like atoms are the fundamental units of matter in science. A prime number is an integer greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, it has exactly two distinct positive divisors: 1 and itself.

Common examples of prime numbers include 2, 3, 5, 7, 11, and so on. In the context of the given problem, we examine Y for primality. Upon careful consideration, we determine that Y, being a product of 17 and a factorial sequence, does not qualify as a prime number because it has more than two factors. This is essential knowledge when evaluating prime factorization or testing the divisibility of numbers for primality.

Factorial Arithmetic

Factorial arithmetic is a branch of mathematics that deals with the multiplication of a series of descending natural numbers. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, the factorial of 5, written as 5!, would be calculated as 5 x 4 x 3 x 2 x 1 = 120.

In our exercise, the factorial concept helps simplify the arithmetic operations involved in calculating X and Y. By recognizing that the series of multiplications forms a part of the factorial sequence, we can condense extensive calculations into a more manageable form, such as 28(1 x 2 x 3 x ... x 16). Understanding factorials is crucial for solving problems related to permutations, combinations, and various areas of discrete mathematics.

Math Problem Solving

Math problem solving is an analytical process where we apply mathematical methods to find solutions to complex problems. It involves several stages, including understanding the problem, devising a plan, executing the plan, and then looking back to review the results. Problem-solving abilities are essential for tackling challenges ranging from simple arithmetic to advanced calculus.

When confronting a math problem, such as the one presented, it's crucial to break down the problem into smaller, more manageable parts. In this case, we first simplified the expressions for X and Y through factorial arithmetic. Next, we evaluated each statement about X and Y's properties, drawing on our knowledge of composite numbers and prime numbers. Through a stepwise and logical approach, we successfully arrived at the correct answer, showcasing effective problem-solving skills.