Which digits should come in place * and $,respectively, if the number 72864*$ is divisible by both 8 and 5?

Correct — C. Divisibility by 5 first: the last digit $ must be 0 or 5.

A multiple of 8 is even, so $ cannot be 5. That fixes $ = 0.

The 8-test reads the last three digits, 4 * 0 = 400 + 10 × *.
400 ÷ 8 = 50 exactly, so 10 × * must itself be a multiple of 8.
That allows * = 0, 4 or 8, and 4 is the value on offer.

Check: 440 ÷ 8 = 55 and 7286440 ÷ 5 = 1457288.
So (*, $) = 4 and 0 — option (c).

Why the others are wrong

• (a)
  4 and 5 — $ = 5 leaves the number odd, and no odd number is divisible by 8. The 5-test passes here, but the 8-test fails before you reach the last three digits.
• (b)
  2 and 0 — $ = 0 clears the 5-test, but the last three digits are then 420, and 420 ÷ 8 = 52.5. The 8-test rejects it.
• (d)
  2 and 5 — Ending in 5 makes the number divisible by 5 but odd, so 8 is out. Both this and option (a) fail on the same parity point.

A number is divisible by 8 when its last three digits form a multiple of 8, and by 5 when its last digit is 0 or 5.

The two tests are independent, so run the cheap one first. The 5-test leaves two candidates for $, and the 8-test — which forces an even number — removes the 5 immediately.

With $ = 0 the last three digits are 400 + 10 × *. Since 400 is already a multiple of 8, the whole question reduces to which * makes 10 × * a multiple of 8.

Key facts

• Divisibility by 8 depends only on the last three digits, because 1000 is a multiple of 8.
• Divisibility by 5 depends only on the last digit, which must be 0 or 5.
• Any number divisible by 8 is even, so a number ending in 5 can never be divisible by 8.
• 7286440 divided by 8 is 910805, and divided by 5 is 1457288.

Common traps

• Testing 8 on the last two digits instead of the last three
• Picking $ = 5 because the 5-test passes, without checking that 8 needs an even number
• Assuming * is unique — 0, 4 and 8 all work here, and the options decide which one is offered

No directly related past PYQ was found.