The ratio of cupcakes to children at a particular birthday party is 104 to 7. Each child at the birthday party eats exactly x cupcakes (where x is a positive integer) and the adults attending the birthday party do not eat anything. If the number of cupcakes that remain uneaten is less than the number of children at the birthday party, what must be true about the number of uneaten cupcakes?
I. It is a multiple of 2.
II. It is a multiple of 3.
III. It is a multiple of 7.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II and III
Assume there are 104 cupcakes and 7 children.
This line: "If the number of cupcakes that remain uneaten is less than the number of children at the birthday party,"
implies that 104 cupcakes are divided equally among 7 children till you get a remainder less than 7.
104/7 => Quotient is 14 (each child gets 14 cupcakes!) and remainder is 6. 6 is a multiple of 2 and 3.
What happens if there are 208 cupcakes and 14 children? (ratio is multiplied by 2)
The quotient will still be 14 and the remainder will be 12 (the remainder will be multiplied by 2)
104a = 7a*14 + 6a
The remainder 6a will always be divisible by 2 and 3.
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