Question

What is the maximum possible value of \(x y\), where \(x+5 \mid=!\) and \(y=9-|x-4|\) (a) 401 (b) 124 (c) 104 (d) none of these

Short answer

Given the equations \(x+5\mid n!\) and \(y=9-|x-4|\), find the maximum possible value of \(xy\). Solution: Step 1: Determine possible values of x We are given that (x+5) divides n!, which means that the factorial of some natural number n is divisible by (x+5). We need to find the possible values of x+5 that can be factors of some n!. Since x+5 must be a factor of n!, x+5 has to be less than or equal to n and the smallest possible value is 1. Step 2: Find the corresponding values of y Now that we have our possible values of x+5, let's find the y values for each of those, using the equation \(y=9-|x-4|\). We can substitute each possible value of x into the given equation and solve for y. Step 3: Calculate the product x * y After finding the corresponding y values for each possible value of x, we need to find the product x * y. By multiplying the values of x and y, we get the product x * y for each pair (x, y). Step 4: Find the maximum value of x * y With the products x * y for all possible pairs (x, y), we simply need to find the maximum value among those products. Compare the products and pick the one with the highest value. This will be the maximum possible value of x * y. After completing these steps, we can determine the maximum possible value of x * y and match it with the given options to see if it matches any of them.

Step-by-step solution

Determine possible values of x

We are given that \((x+5)\mid!\). This means that the factorial of some natural number is divisible by \(x+5\). Let's assume this natural number is \(n\). Then, \(n!\) is a multiple of \((x+5)\). Hence, \(x+5\) must be a factor of \(n!\). Now, we need to find all possible values of \(x+5\) that can be factors of some \(n!\). Remember, since\(x+5\mid n!\), \(x+5\leq n\) and the smallest possible value is 1.

Find the corresponding values of y

Now that we have our possible values of \(x+5\), let's find the y values for each of those, using the equation \(y=9-|x-4|\). One way to obtain the corresponding values of \(y\) is to substitute each possible value of \(x\) into the given equation and solve for \(y\).

Calculate the product x * y

After finding the corresponding y values for each possible value of x, we need to find the product \(x \cdot y\). By multiplying the values of x and y, we get the product \(x \cdot y\) for each pair \((x, y)\).

Find the maximum value of x * y

Now that we have the products \(x \cdot y\) for all possible pairs \((x, y)\), we simply need to find the maximum among those products. Compare the products and pick the one which has the highest value. This will be the maximum possible value of \(x\cdot y\). After completing these steps, we can determine the maximum possible value of \(x \cdot y\), and match it with the given options to see if it matches any of them.

Key concepts

Factorials

In mathematics, factorial is a function that multiplies a series of descending natural numbers. For any natural number \( n \), its factorial is represented by \( n! \) and is calculated as \( n! = n \times (n-1) \times (n-2) \times \, ... \, \times 2 \times 1 \).

Factorials are very helpful in problems involving permutations, combinations, and various algebraic expressions. In our exercise, the expression \((x+5)\mid n!\) means that the factorial of \( n \) should be divisible by \( x+5 \). This ties factorials to divisibility, highlighting that each value of \( x+5 \) should be a divisor of a factorial of some natural number \( n \).

To find if a number is a factor of \( n! \), consider the numbers up to \( n \). For instance, the value 5 is part of \( n! \) if \( n \geq 5 \). Factorials increase very rapidly, so often only smaller numbers will be factors of a factorial.

Algebraic Expressions

Algebraic expressions involve variables and constants combined using operators like addition, subtraction, multiplication or division. Consider the expression \( y = 9 - |x - 4| \), which includes a variable \( x \), the constant 9, and a subtraction operator applied to an absolute value.

Absolute value \( |x - 4| \) captures how far \( x \) is from 4, without regard to direction along the number line. The greater \( x \) deviates from 4, the smaller \( y \) becomes. This algebraic manipulation indicates that when choosing an \( x \), ones that bring it closer to 4 result in a larger value of \( y \).

Solving such expressions requires you to separate positive and negative possibilities:

• If \( x > 4 \), then \(|x-4| = x-4\).
• If \( x < 4 \), then \(|x-4| = 4-x\).

These variations help effectively solve while ensuring all conditions are met for each possible value of \( x \).

Problem Solving Steps

Problem solving often involves breaking down a complex task into simpler, manageable stages. In the given exercise, finding the maximum product of \( x \cdot y \) required several steps, each systematically uncovering necessary details.

Step 1 involves determining the potential values of \( x \) tied to the constraint \((x+5)\mid n!\). This equation establishes the range of \( x \) values that satisfy \((x+5)\) as factors of factorials of certain \( n \).

In Step 2, compute \( y \) for each \( x \) using \( y = 9 - |x - 4| \). Here, every allowable \( x \) yields associated \( y \) values. Calculating exact values demands understanding the functional relationship within the equation, especially concerning absolute values.

Step 3 directs you to calculate \( x \cdot y \), where each \( (x, y) \) pair leads to a distinct product. It's often insightful to use a strategic comparison of these different products to understand potential outcomes.

Finally, in Step 4, finding the largest \( x \cdot y \) involves selecting the highest product from earlier computations. Extracting the maximum value aligns with choosing the answer from provided options, reflecting both computed knowledge and logical deduction.