How many 3-digit natural numbers are divisible by 6?

Given data

• 3-digit numbers range from 100 to 999.
• Divisible by 6 ⇔ divisible by 2 and 3.

Concept / Approach

• Count multiples of 6 in an inclusive interval using arithmetic progression logic.

Step-by-step calculation

Smallest 3-digit multiple of 6: 102 (since 100 is not; 102 = 6×17)Largest 3-digit multiple of 6: 996 (since 999 is not; 996 = 6×166)Count = ((996 − 102)/6) + 1 = (894/6) + 1 = 149 + 1 = 150

Alternative formula

Count = ⌊999/6⌋ − ⌊99/6⌋ = 166 − 16 = 150.

Common pitfalls

• Accidentally including 100 or 999.
• Forgetting the +1 when using the AP term-count formula.

Final Answer

150