Divisibility by 3 and 9: add up the digits

The two rules

• Divisible by 3: the sum of digits is divisible by 3.
• Divisible by 9: the sum of digits is divisible by 9.

In one picture

Take $471$:

• Sum of digits: $4 + 7 + 1 = 12$.
• $12$ is divisible by 3 → 471 is divisible by 3 ✓ (check: $471 = 3 \times 157$).
• $12$ isn’t divisible by 9 → 471 isn’t divisible by 9.

Take $342$:

• $3 + 4 + 2 = 9$ → divisible by 3 and 9 ✓ (check: $342 = 9 \times 38$).

The shortcut of the shortcut

If the sum is still big, sum again. Example $918273$:

• $9 + 1 + 8 + 2 + 7 + 3 = 30$
• $3 + 0 = 3$ → divisible by 3, not by 9.

Why it works

Every power of 10 equals $1$ modulo 9 (because $10 = 9 + 1$). So a number is congruent to the sum of its digits modulo 9. Divisible by 9 = digit sum divisible by 9. Same for 3 since $9 = 3 \times 3$.

?Your turn

Is 2718 divisible by 9?

A.YesB.No

Check

?Your turn

Among these 4 numbers, which one is NOT divisible by 3?

A.1023B.4567C.8190D.3333

Check