how many 3-digit numbers are completely divisible 6 ?

All 3-digit numbers that are completely divisible by 6 are multiples of 6 from 102 to 996, and there are 150 such numbers.

Here’s a quick way to see it:

• Smallest 3-digit number is 100.
  • First multiple of 6 ≥ 100 is 102.

• Largest 3-digit number is 999.
  • Last multiple of 6 ≤ 999 is 996.

These form an arithmetic progression: 102, 108, 114, ..., 996 with common difference 6.

Use the nth-term formula l=a+(n−1)dl=a+(n-1)dl=a+(n−1)d (last term = first term + (n − 1) × common difference):

• 102102102 is the first term aaa
• 996996996 is the last term lll
• d=6d=6d=6

996=102+(n−1)⋅6996=102+(n-1)\cdot 6996=102+(n−1)⋅6

996−102=(n−1)⋅6996-102=(n-1)\cdot 6996−102=(n−1)⋅6

894=(n−1)⋅6894=(n-1)\cdot 6894=(n−1)⋅6

n−1=149,n=150n-1=149,\quad n=150n−1=149,n=150

So, 150 three-digit numbers are completely divisible by 6.