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Explanation:

To find remainder of any number when divided by 8, we need to find remainder of last 3 digits of the number when divided by 8.

Here we need to find the last 3 digits of 1223334444… (500 digits)

The given number is formed by writing consecutive natural numbers as many times.

1 is written once, 2 is written twice, 3 is written thrice and so on.

By the time number 1 is over, only 1 digit is written.
By the time number 2 is over, (1 + 2 =) 3 digits are written.
By the time number 3 is over, (1 + 2 + 3 =) 6 digits are written.

By the time all single digit numbers are written, (1 + 2 + 3 + … + 9 =) 45 digits are written.

By the time number 11 is over, 45 + 2 × 11 = 67 digits are written.

By the time number 23 is over, 45 + 2 × (11 + 12 + 13 + … + 23 =) 487 digits are written.

Now thirteen digits are remaining which will be formed by writing 24 repeatedly.

The last three digits of this number will be 242.

∴ The last 3 digits of the given number is 242.

Remainder when 242 is divided by 8 = 2.

∴ Remainder when the given number is divided by 8 = 2.

Hence, 2.

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