When a number is divided by a certain divisor, the remainder is 46. If another number is divided by same divisor, it leaves remainder 31. If the sum of both the numbers is divided by the same divisor, it gives remainder 19. Find divisor

Let's denote the unknown divisor as D.

1. From the first condition, we can write the first number as aD + 46, where 'a' is the quotient when the first number is divided by D.

2. From the second condition, we can write the second number as bD + 31, where 'b' is the quotient when the second number is divided by D.

3. According to the problem, the sum of the two numbers gives a remainder of 19 when divided by D. So, we can write this as: (aD + 46) + (bD + 31) = cD + 19, where 'c' is the quotient when the sum of the two numbers is divided by D.

4. Simplifying the above equation gives us: aD + bD + 77 = cD + 19.

5. Rearranging terms, we get: (a + b - c)D = 19 - 77.

6. Simplifying further, we get: (a + b - c)D = -58.

7. Since D is a divisor, it must be a positive number. Also, (a + b - c) must be a positive number because they represent quotients which are always positive. But the right side of the equation is negative, which is a contradiction.

8. Therefore, there seems to be a mistake in the problem as it stands, because it leads to an impossible situation. The divisor D cannot be found with the given conditions.