Can we conclude that as the ratio is always 104 to 7, the remainder will be 6 for any no. x and hence D?

Hey,

Yes the ratio will be the same. So you can take values as well.
Considering : The number of cupcakes that remain uneaten is less than the number of children at the birthday party

Take cupcakes $= 104$ & no of children = $7$ - Uneaten cupcakes = $6$
Or cupcakes = $208$ & no of children = $14$ - Uneaten cupcakes = $12$

Hence the no of uneaten cupcakes will always be divisible by 2 & 3.

Please ask if the doubt remains.

The correct answer is (D), I and II only. Since the ratio of cupcakes to children is 104 to 7, and each child eats x cupcakes, the total number of cupcakes must be 104x. If the number of uneaten cupcakes is less than the number of children, then the number of uneaten cupcakes must be 104x - 7. Since 104x - 7 is a multiple of 7, the answer must be III. However, since 104x - 7 is also a multiple of 2 and 3, the answer must also be I and II. Therefore, the correct answer is (D), I and II only.

I hope this answer was helpful! If you need any more essay help UK, feel free to reach out.

104 - 7x < 7
Carcass i was able to get this far from the stem, pls help for the next step, above explanations are flying over my head

Given that:
The ratio of cupcakes to children is 104 to 7 --> $\frac{cupcakes}{children}=\frac{104k}{7k}$;
Each child eats exactly x cupcakes --> the number of cupcakes eaten $7kx$ and the number of cupcakes that remain uneaten is $104k-7kx$;
The number of cupcakes that remain uneaten is less than the number of children --> $104k-7kx<7k$ --> $x>13\frac{6}{7}$ --> $x=14$ (notice that x cannot be more than 14 since in this case 7kx>104k, which would mean that more cupcakes were eaten than there were).
Now, if $x=14$, then the number of cupcakes that remain uneaten is $104k-7k*14=6k$, thus the number of uneaten cupcakes must be a multiple of both 2 and 3.
Answer: D.

let me know if now is clear