How many of these are divisible by 3 but not by 9: 2133, 2343, 3474, 4131, 5286, 5340, 6336, 7347, 8115, 9276?

Given data

• We must count numbers divisible by 3 but not by 9.

Concept / Approach

• Divisible by 3 ⇔ digit sum is a multiple of 3.
• Divisible by 9 ⇔ digit sum is a multiple of 9.
• So we want digit sum ≡ 0 (mod 3) but ≠ 0 (mod 9).

Check each number

2133: sum = 9 ⇒ divisible by 9 ⇒ exclude.2343: sum = 12 ⇒ /3 yes, /9 no ⇒ include.3474: sum = 18 ⇒ /9 yes ⇒ exclude.4131: sum = 9 ⇒ /9 yes ⇒ exclude.5286: sum = 21 ⇒ /3 yes, /9 no ⇒ include.5340: sum = 12 ⇒ /3 yes, /9 no ⇒ include.6336: sum = 18 ⇒ /9 yes ⇒ exclude.7347: sum = 21 ⇒ /3 yes, /9 no ⇒ include.8115: sum = 15 ⇒ /3 yes, /9 no ⇒ include.9276: sum = 24 ⇒ /3 yes, /9 no ⇒ include.

Total

Included: {2343, 5286, 5340, 7347, 8115, 9276} ⇒ count = 6.

Common pitfalls

• Assuming “divisible by 3” automatically means “divisible by 9.” It does not.

Final Answer

6