What is the remainder when 987654321 is divided by 9?

Correct — C.
Rule: a number leaves the same remainder on division by 9 as the sum of its digits.

Digit sum: 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45
45 = 9 × 5, remainder 0
Check: 9 × 109739369 = 987654321

Remainder 0 → option (c).

Why the others are wrong

• (a)
  1 — 1 would need a digit sum one more than a multiple of 9, such as 46. The digits add to 45, which is exactly 9 × 5.
• (b)
  3 — 3 would need a digit sum three more than a multiple of 9, such as 48. The digits of 987654321 add to 45, which leaves nothing over.
• (d)
  9 — 9 cannot be a remainder on division by 9 at all: the possible remainders are 0 to 8. A digit sum that divides exactly means remainder 0.

Divisibility by 9 is tested with the digit sum. Every power of 10 leaves remainder 1 on division by 9 (10 = 9 + 1, 100 = 99 + 1), so each digit adds its own value to the remainder. The digits' total therefore leaves the same remainder as the number.

Here the digits 1 to 9 add to 45, a multiple of 9, so the remainder is 0.

Key facts

• The digits 1 to 9 add up to 45.
• A number is divisible by 9 exactly when its digit sum is divisible by 9.
• Division by 9 leaves a remainder from 0 to 8.

Common traps

• Choosing 9 as the remainder when the digit sum is a multiple of 9
• Dropping one of the nine digits while adding them

No directly related past PYQ was found.