Given
N= 2 mod 3
N=0 mod 5

Hence, N=5 mod 15
or N= 15k+5

k can take 6 values.

N=3x+2
N=5y
N is a 2 digit positive integer
10≤5y<100
2≤y<20

3x+2=5y
3x=5y-2
x=(5y-2)/3
As x and y are integers, y can be 4, 7,10, 13, 16 or 19 (only these values of y lie in highlighted range).

from given info
we know that 3x+2=5y
x=5y-2/3 ; so possible integer value of y & x will be when y = 4,7,10,13,16 ,19
total 6 possible cases
IMO B

Rumsus ,
Let me rephrase the question for you.
The number of gift boxes when divided among THREE people gives a remainder of 2.
The number of gift boxes when divided among FIVE people gives a remainder of 0.
The number of gift boxes is a two digit number.
Find how many such two digit numbers exist?

First point:
The number of gift boxes when divided among THREE people gives a remainder of 2.

2,5,8,11...17,20 23...35 ..50....65...80...95...101..104
All the above numbers when divided by 3 give a remainder of two, but we are only interested in such two digit numbers.(Bold ones)

Second point:
The number of gift boxes when divided among FIVE people gives a remainder of 0

5,10,15, 20,25,30,...35 ..50....65...80...95.......100....105
All the above numbers when divided by 5 give a remainder of zero, but we are only interested in such two digit numbers and which are common with the list above(Bold ones)


The number of gift boxes common in both lists above and satisfying the given criteria.
20 35 ..50....65...80...95

Hence there are six such numbers possible. Hope now it's is clear why we are taking the second point to be a multiple of 5 ( 5K).
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