What digit x makes the number 4x6723 divisible by 11?
x = 2. For divisibility by 11, the difference between the sum of digits in odd places and in even places (counted from the right) must be 0 or a multiple of 11. Here that difference is x - 2, so x = 2, giving 426723 = 11 x 38793.
Explain it simply
The rule for 11 is like a tug of war between two teams of digits. Number the places from the right. Digits in places 1, 3, 5 pull one way; digits in places 2, 4, 6 pull the other way. If the two teams are perfectly balanced (difference 0) or differ by exactly 11, 22 and so on, the number is divisible by 11.
Example
Question: 4x6723 is divisible by 11. Find the digit x.
Digits from the right: 3 (place 1), 2 (place 2), 7 (place 3), 6 (place 4), x (place 5), 4 (place 6).
- Odd places: 3 + 7 + x = 10 + x.
- Even places: 2 + 6 + 4 = 12.
- Difference: (10 + x) - 12 = x - 2.
- x is a single digit (0 to 9), so x - 2 can only be 0 (it cannot reach 11 or -11). So x = 2.
Check: 426723 / 11 = 38793 exactly.
console.log(426723 % 11); // Output: 0What to say in the interview
"I use the alternating-sum rule for 11: odd-place digits minus even-place digits must be a multiple of 11, including zero. That gave x - 2 = 0, so x is 2, and I confirmed by dividing."
Common mistakes
- Forgetting that a difference of 11 or -11 also works, which matters when the digit sums are large.
- Counting places from the left in one step and from the right in another. Either direction works, as long as you stay consistent.