To find the smallest possible integer when divided by 3, 4, or 5 with a remainder of 2, we need to find the least common multiple (LCM) of these three numbers and then add 2 to that number.
The LCM of 3, 4, and 5 is 60. To understand this, we can find the LCM by examining the multiples of each number:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60...
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60...
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60...
We can see that the smallest number that satisfies the remainder of 2 when divided by 3 is 2, when divided by 4 is 4, and when divided by 5 is 5. The first number that satisfies all three conditions is 60.
Finally, we add 2 to the LCM (60 + 2) to get the smallest possible integer when divided by 3, 4, or 5 with a remainder of 2, which is 62.
Therefore, the smallest possible integer that meets the given conditions is 62.