Understanding the Problem: Divisibility by 11

The question asks us to find the face value of the digit 'x' in the number x3331, given that this number is exactly divisible by 11. To solve this, we need to use the divisibility rule for 11.

Divisibility Rule of 11 Explained

A number is divisible by 11 if the difference between the sum of its digits at odd places (starting from the rightmost digit, the unit's place) and the sum of its digits at even places is either 0 or a multiple of 11.

Let's break down the number x3331 based on the positions of its digits, starting from the right:

• 1 is at the 1st place (odd)
• 3 is at the 2nd place (even)
• 3 is at the 3rd place (odd)
• 3 is at the 4th place (even)
• x is at the 5th place (odd)

Applying the Divisibility Rule to x3331

Now, let's calculate the sum of digits at odd places and the sum of digits at even places for the number x3331.

• Sum of digits at odd places (1st, 3rd, 5th) = Digit at 1st place + Digit at 3rd place + Digit at 5th place
• Sum of digits at odd places = \(1 + 3 + x = 4 + x\)

• Sum of digits at even places (2nd, 4th) = Digit at 2nd place + Digit at 4th place
• Sum of digits at even places = \(3 + 3 = 6\)

Next, we find the difference between these two sums:

Difference = (Sum of digits at odd places) - (Sum of digits at even places)

Difference = \((4 + x) - 6\)

Difference = \(x - 2\)

Finding the Face Value of x

According to the divisibility rule of 11, the number x3331 is divisible by 11 if this difference \((x - 2)\) is either 0 or a multiple of 11.

Since x is a single digit (a face value), its possible values are 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Let's consider the range of values for the difference \(x - 2\):

• If \(x = 0\), difference = \(0 - 2 = -2\)
• If \(x = 1\), difference = \(1 - 2 = -1\)
• If \(x = 2\), difference = \(2 - 2 = 0\)
• If \(x = 3\), difference = \(3 - 2 = 1\)
• ...
• If \(x = 9\), difference = \(9 - 2 = 7\)

The possible values for the difference \(x - 2\) range from -2 to 7.

We need the difference to be 0 or a multiple of 11. The only multiple of 11 within the range of -2 to 7 is 0.

Therefore, we must have:

\(x - 2 = 0\)

Solving for x:

\(x = 2\)

The face value of x must be 2.

Verification

Let's check if the number 23331 is divisible by 11 using the rule:

• Sum of digits at odd places (1st, 3rd, 5th): \(1 + 3 + 2 = 6\)
• Sum of digits at even places (2nd, 4th): \(3 + 3 = 6\)
• Difference = \(6 - 6 = 0\)

Since the difference is 0, the number 23331 is indeed divisible by 11.

This confirms that the face value of x is 2.

Conclusion on Face Value of x

Based on the divisibility rule of 11, the face value of x in the number x3331 must be 2 for the number to be divisible by 11.

Revision Table: Key Concepts

Additional Information: Divisibility Rules in Number Theory

Divisibility rules are shortcuts used to determine if a number is divisible by another number without performing long division. They are fundamental concepts in number theory and are often useful in solving problems involving factors, multiples, and prime numbers.

Other common divisibility rules include:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Divisibility by 10: A number is divisible by 10 if its last digit is 0.

Understanding these rules can significantly speed up calculations and problem-solving in various mathematical contexts.