7. Find the reminder when 333666777888999 is divided by 3

or 9 or 11 ?

3-0

9-0
11-5

Add up the digits - take mod 9. That's the remainder mod 9

Add up the digits - take mod 3. That's the remainder mod 3

Go from left to right, alternately subtracting and adding digits. The result is the remainder mod 11
(This is because 100 = 1 mod 11 but 10 = -1 mod 11)

To find the remainder when a number is divided by another number, we can use the remainder theorem. According to the remainder theorem, if we divide a number by another number, the remainder will be the same as the remainder when we divide the sum of its digits by the given number.

To find the remainder when 333666777888999 is divided by 3:
Step 1: Find the sum of the digits: 3 + 3 + 3 + 6 + 6 + 6 + 7 + 7 + 7 + 8 + 8 + 8 + 9 + 9 + 9 = 99.
Step 2: Divide the sum (99) by 3.
Since 99 is divisible by 3 without any remainder, the remainder when 333666777888999 is divided by 3 is 0.

To find the remainder when 333666777888999 is divided by 9:
Step 1: Find the sum of the digits: 3 + 3 + 3 + 6 + 6 + 6 + 7 + 7 + 7 + 8 + 8 + 8 + 9 + 9 + 9 = 99.
Step 2: Divide the sum (99) by 9.
Since 99 is divisible by 9 without any remainder, the remainder when 333666777888999 is divided by 9 is 0.

To find the remainder when 333666777888999 is divided by 11:
Step 1: Alternate between adding and subtracting the digits from left to right: 3 - 3 + 3 - 6 + 6 - 6 + 7 - 7 + 7 - 8 + 8 - 8 + 9 - 9 + 9 = 0.
Step 2: If the difference obtained in step 1 is divisible by 11, then the original number is also divisible by 11. In this case, the difference is 0, which is divisible by 11.
Since the difference (0) is divisible by 11 without any remainder, the remainder when 333666777888999 is divided by 11 is also 0.