Multiple choice

If a certain number is divided by the sum of its two digits, the quotient is $6$ and remainder is $3$. If the digits are interchanged and the resulting number is divided by the sum of the digits, then the quotient is $4$ and the remainder is $9$. Find the number.

  1. $50$
  2. $75$
  3. $65$
  4. $35$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the number be 10x + y. (10x + y) = 6(x + y) + 3 => 4x - 5y = 3. (10y + x) = 4(x + y) + 9 => -3x + 6y = 9 => -x + 2y = 3. Solving: x = 2y - 3. 4(2y - 3) - 5y = 3 => 8y - 12 - 5y = 3 => 3y = 15 => y = 5. x = 2(5) - 3 = 7. Number is 75.

AI explanation

Let the number be represented as 10a plus b. Using the division algorithm, the first condition gives 10a plus b equals 6 multiplied by (a plus b) plus 3, which simplifies to 4a equals 5b plus 3. The second condition gives 10b plus a equals 4 multiplied by (a plus b) plus 9, which simplifies to 6b equals 3a plus 9. Solving this system of equations, we find a equals 7 and b equals 5. Therefore, the number is 75.

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