Understanding the Divisibility Problem for 247xy

The problem states that a five-digit number, 247xy, is divisible by 3, 7, and 11. We need to find the value of the expression \((2y - 8x)\), where x and y are the digits in the units and tens place, respectively.

For a number to be divisible by 3, 7, and 11, it must be divisible by the Least Common Multiple (LCM) of these three numbers. Since 3, 7, and 11 are all prime numbers, their LCM is simply their product.

Let's calculate the LCM of 3, 7, and 11:

\(\text{LCM}(3, 7, 11) = 3 \times 7 \times 11 = 21 \times 11 = 231\)

So, the number 247xy must be a multiple of 231.

Finding the Five-Digit Number 247xy

The number 247xy can be written in expanded form as \(24700 + 10x + y\). Since x and y are digits, x and y can range from 0 to 9.

• If x=0 and y=0, the number is 24700.
• If x=9 and y=9, the number is 24799.

Thus, the number 247xy is a multiple of 231 that lies in the range from 24700 to 24799, inclusive.

We need to find a multiple of 231 within this range. Let's find multiples of 231 around 24700:

• We can estimate by dividing 24700 by 231: \(24700 \div 231 \approx 106.92\).
• This means the multiples of 231 near 24700 are \(231 \times 106\), \(231 \times 107\), \(231 \times 108\), and so on.
• Let's calculate these multiples:
  • \(231 \times 106 = 24486\) (This is less than 24700, so it's not the number).
  • \(231 \times 107 = 24717\) (This is within the range 24700-24799).
  • \(231 \times 108 = 24948\) (This is greater than 24799, so it's not the number).

The only multiple of 231 between 24700 and 24799 is 24717.

Therefore, the five-digit number 247xy is 24717.

Determining the Values of x and y

By comparing 247xy with 24717, we can determine the values of x and y:

• The digit in the tens place, x, is 1.
• The digit in the units place, y, is 7.

So, \(x = 1\) and \(y = 7\).

Calculating the Value of (2y - 8x)

Now that we have the values of x and y, we can calculate the value of the expression \((2y - 8x)\):

\(2y - 8x = 2(7) - 8(1)\)

\(2y - 8x = 14 - 8\)

\(2y - 8x = 6\)

The value of \((2y - 8x)\) is 6.

Revision Table: Key Concepts

Additional Information: Divisibility Rules

Here are some basic divisibility rules that are often useful:

• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 7: To check if a number is divisible by 7, subtract twice the last digit from the number formed by the remaining digits. Repeat this process until you get a small number. If this number is 0 or divisible by 7, the original number is divisible by 7.
• Divisibility by 11: A number is divisible by 11 if the alternating sum of its digits (starting from the rightmost digit) is 0 or divisible by 11. For example, for a number abcde, the alternating sum is \(e - d + c - b + a\).

In our case, for 24717:

• Sum of digits: \(2 + 4 + 7 + 1 + 7 = 21\). Since 21 is divisible by 3, 24717 is divisible by 3.
• Divisibility by 7:
  • \(2471 - 2 \times 7 = 2471 - 14 = 2457\)
  • \(245 - 2 \times 7 = 245 - 14 = 231\)
  • \(23 - 2 \times 1 = 23 - 2 = 21\)
  Since 21 is divisible by 7, 24717 is divisible by 7.
• Divisibility by 11: Alternating sum: \(7 - 1 + 7 - 4 + 2 = 6 + 3 + 2 = 11\). Since 11 is divisible by 11, 24717 is divisible by 11.

These checks confirm that 24717 satisfies all the given conditions.