If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 9.
Step 1: Recall two rules.
• Test for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Test for 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
Step 2: The statement says: “Sum divisible by 3 ⇒ number divisible by 9.” We will check this with a counterexample.
Step 3 (Counterexample): Take the number (12).
Digit sum: (1 + 2 = 3).
(3) is divisible by (3) ⇒ (12) is divisible by (3).
Now check divisibility by (9): compute (12 div 9).
(12 div 9 = 1) remainder (3) ⇒ (12) is not divisible by (9).
Step 4 (Conclusion): Having the digit sum divisible by (3) only guarantees divisibility by (3), not by (9). Therefore, the statement is false.
Note: Every multiple of (9) is also a multiple of (3), but not every multiple of (3) is a multiple of (9).