When a positive integer x is divided by d, the remainder is 241 and

"when a pos. integer x is divided by divisor d, the remainder is 241"

given: (x / d) = Q + (241 / d)
or, the positive integer x can be written as:

x = d(Q) + 241

Property: the remainder must always be less than the divisor

so, divisor d > 241 ---- (we can eliminate answer (A) because it is too small)

"when 2x is divided by d, the remainder is 112"

when we DOUBLE the positive integer x, we are doubling the representation of the value.
let X = the same value represented as X number of balls that we have on the floor

originally, we were able to take X and apportion the balls into Q-quotient groups of D.
however, there were 241 balls left over that could not be made into a group of D balls.

if we take that same picture and DOUBLE it, we will have 241 and 241 balls from each that we were unable to make into a group of D balls.

however, we are told the remainder is 112.

this means we were able to take these (241 + 241 = 482 balls) and make Another group of D balls, with 112 left over.

this also tells us that Divisor - D must be less than < 482 ----we can eliminate answer (E) right away

we can take these 482 balls, divide them into one D group, and have a remainder of 112:

482 = D(1) + 112

D = 370

answer (D)

Tried an alternate way, let me know if this helps!

We are given positive integer X when divided by d gives remainder as 241.
We can form the remainder equation as X = d * q1 + 241 ......(d is the divisor, q1 is quotient)

Next stem says when 2X is divided by d gives remainder as 112.
Remainder equation: 2X = d * q2 + 112

= 2X = 2 * X
= d * q2 + 112 = 2* (d * q1 + 241) ....... (Substituting from remainder equations)
= d * (q2 - 2 * q1) = 482 - 112
= d * (q2 - 2 * q1) = 370

Drawing some observations to derive d,
1. d and (q2 - 2 * q1) are either both negative or both positive since +370. I'm eliminating negative possibilities since d is positive in answer choices.
2. In remainder equation, q1 and q2 are integers, so (q2 - 2 * q1) is a positive integer.

So, d * (+ve integer k) = 370.
Putting in values, k=1, d=370
k=2, d=185
k=5, d=72

Only k=1 fits the bill, thus d=370. Option D.

PS: Remainder equation: Number = Divisor * Quotient + Remainder.
eg. 73 = 11 * 6 + 7

Mathematically :

X = d*a + 241 ( a some random multiple )
2X = d*b + 112 ( b some random multiple )

solving theses we get:

2d*a - d*b = 112 - 482
d * (2a - b) = 370

now 37*2*5 can be broken into two integers for form d * (2a - b) like:
370 * 1
185 * 2
74 * 5

Since n > 241 only n = 370 fits our description.

How do we judge that divisor is greater than dividend