Which of the following numbers is divisible by 2, 5, 10 and 11?

Correct — C. Divisibility by 2, 5 and 10 together collapses into a single test: the number must end in 0.

203467 ends in 7 and 830942 ends in 2, so both fail at 5 and 10. That leaves 589270 and 1234560.

Now the 11-test, taking the alternating sum of the digits from the right.
589270 → 0 − 7 + 2 − 9 + 8 − 5 = −11, a multiple of 11.
1234560 → 0 − 6 + 5 − 4 + 3 − 2 + 1 = −3, not a multiple of 11.

589270 clears all four divisors → option (c).

Why the others are wrong

• (a)
  203467 — 203467 ends in 7, so it is neither even nor a multiple of 5. Three of the four required divisors fail on the last digit alone.
• (b)
  830942 — 830942 ends in 2, so it clears 2 but not 5 and not 10. No further testing is needed.
• (d)
  1234560 — 1234560 passes 2, 5 and 10 easily, but its alternating digit sum is −3, so 11 does not divide it. This is the option built to survive the easy screen and fail the last test.

A number divisible by both 2 and 5 is divisible by their LCM, 10, so all three of those conditions reduce to one: a last digit of 0. That screen removes half the options before any real work.

The remaining condition is divisibility by 11, tested by the alternating sum of digits: add and subtract the digits in turn and check whether the result is 0 or a multiple of 11. The direction does not matter as long as the signs alternate consistently.

Screening first and testing second is the time-saving habit this item rewards.

Key facts

• A number is divisible by 10 exactly when its last digit is 0, which also settles 2 and 5.
• Divisibility by 11: the alternating sum of the digits is 0 or a multiple of 11.
• 589270 = 2 × 5 × 11² × 487, so 110 divides it.
• 1234560 is divisible by 2, 3, 4, 5, 8 and 10, but its alternating sum −3 rules out 11.

Common traps

• Running the 11-test on all four options before screening on the last digit.
• Letting the signs slip mid-way through the alternating sum, which is easy on a seven-digit number.
• Picking 1234560 because it looks divisible by everything.

No directly related past PYQ was found.