Final answer:
The remainder when the number is divided by 84, we examine the consistency of remainders from divisions by 3, 4, and 7. The number that fits the given criteria will have the same remainder of 1 when divided by 84, because it's the LCM of 3, 4, and 7.
Step-by-step explanation:
When a number is divided by 3, 4, and 7, the remainders are 2, 1, and 4, respectively. To find out what the remainder would be when this number is divided by 84, we need to understand that 84 is the least common multiple (LCM) of 3, 4, and 7.
However, in this scenario, we need to find a number that, when divided by 3, 4, and 7, gives us the remainders mentioned above. For such a number to exist, the remainders when divided by the LCM (84) must be consistent with the remainders given by dividing by 3, 4, and 7.
Since the remainders of dividing by 3, 4, and 7 are respectively 2, 1, and 4, we can write the number as:
Where k, m, and n are integers. Since this number must leave the same remainder when divided by 84, and because the LCM of 3, 4, and 7 is 84, we look for a remainder that, when added to multiples of 84, still gives remainders of 2, 1, and 4 when divided by 3, 4, and 7 respectively.
In this case, that remainder is 1. So, when the number is divided by 84, the remainder is 1.