Find the least value of x for which 57x716 is divisible by 9.

Correct — D. A number is divisible by 9 exactly when its digit sum is divisible by 9.

Digit sum = 5 + 7 + x + 7 + 1 + 6
= 26 + x
26 leaves remainder 8 on division by 9, so x has to supply the missing 1.

x = 1 → sum = 27 = 9 × 3
The next value that would work is 10, which is not a digit, so 1 is both the least and the only digit that fits — option (d).

Why the others are wrong

• (a)
  8 — With x = 8 the digit sum is 34, which sits between 27 and 36 and so is not a multiple of 9. Even if it did work, 8 could not be the least digit available.
• (b)
  9 — With x = 9 the digit sum is 35 — one short of 36. Picking 9 also reads the stem backwards: it asks for the least value, not the largest digit.
• (c)
  0 — With x = 0 the digit sum is 26, not a multiple of 9. Zero is the reflex answer to any least-value stem, but it still has to be tested before it is chosen.

Divisibility by 9 is a digit-sum test: N is divisible by 9 exactly when the sum of its digits is.

It works because 10 leaves remainder 1 on division by 9, so every power of ten does too, which makes a number and its digit sum leave the same remainder.

With one digit unknown, add the known digits, take the remainder mod 9, and the missing digit is whatever completes the next multiple of 9. Here 26 needs 1 to reach 27.

Key facts

• A number is divisible by 9 when the sum of its digits is divisible by 9.
• The digits of 57x716 other than x sum to 26, and the next multiple of 9 above 26 is 27.
• The same digit-sum test with 3 in place of 9 decides divisibility by 3.
• Divisibility by 11 uses the alternating digit sum, not the plain digit sum.

Common traps

• Reading least value as smallest option printed and choosing 0 without testing it
• Choosing the digit equal to the remainder instead of the digit that completes the next multiple of 9
• Applying a digit-sum test to 8 or 4, where only the trailing digits matter

No directly related past PYQ was found.