Question

The sum of the digits of a four digit positive integer is 14. its two middle digit have a sum of nine and the thousands digit minus the units digit is one. if the integer is divisle by 11. what is the integer

Answer 1

The four-digit positive integer is 3452.

Step-by-step explanation:

The given information states that the sum of the digits of a four-digit positive integer is 14. Let's represent the thousands digit as A, the hundreds digit as B, the tens digit as C, and the units digit as D. From the information, we can set up the following equations:

1. A + B + C + D = 14
2. B + C = 9
3. A - D = 1

We also know that the integer is divisible by 11. A number is divisible by 11 if the difference between the sum of its odd-placed digits and the sum of its even-placed digits is divisible by 11. In this case, since there are four digits, we need to consider C as the odd-placed digit and B as the even-placed digit. The equation for divisibility by 11 is:

C - B ≡ 0 (mod 11)

From equation 2, we can express B in terms of C as B = 9 - C. Substituting this into equation 4, we get:

C - (9 - C) ≡ 0 (mod 11)

Simplifying the equation, we have:

2C - 9 ≡ 0 (mod 11)

2C ≡ 9 (mod 11)

By trial and error, we find that C = 5 satisfies this equation. Substituting C = 5 into equation 2, we get B = 9 - C = 9 - 5 = 4. Substituting A = D + 1 into equation 1, we have:

(D + 1) + 4 + 5 + D = 14

2D + 10 = 14

2D = 4

D = 2

Finally, substituting C = 5, B = 4, and D = 2 into the original representation of the integer, we get:

A B C D = 3 4 5 2