If the 9-digit number 5x79856y6 is divisible by 36, then what is the negative value of √(2x + y) for the largest possible value of y, given x and y are natural numbers?

Correct — A. The stem is printed as an image: the 9-digit number 5x79856y6 is divisible by 36, and you want the negative value of √(2x + y) for the largest possible y.

36 = 4 × 9, and 4 and 9 share no factor, so the number must pass both tests.

Divisibility by 4 reads only the last two digits, y6. Of 06, 16, 26 … 96, the multiples of 4 are 16, 36, 56, 76 and 96 — so y is odd.

Divisibility by 9 reads the digit sum. The fixed digits give 5 + 7 + 9 + 8 + 5 + 6 + 6 = 46, so the sum is 46 + x + y. Since 46 leaves remainder 1 on division by 9, x + y = 8 or 17.

The largest odd y is 9, and only the branch x + y = 17 can carry it: x = 8.
Check 587985696 — digit sum 63, last two digits 96. Both tests pass.

2x + y = 16 + 9 = 25
√25 = 5, and the question asks for the negative value → −5, option (a).

Why the others are wrong

• (b)
  −4 — −4 needs 2x + y = 16. But y must be odd for y6 to be a multiple of 4, and 2x + an odd number is always odd — so 2x + y can never equal 16.
• (c)
  −2 — −2 needs 2x + y = 4, which fails on the same parity ground and is far too small anyway: y alone reaches 9.
• (d)
  −7 — −7 needs 2x + y = 49. Even at the largest y = 9 that forces x = 20, and x is a single digit, so no such number exists.

A divisibility test for a composite number is built by splitting it into coprime factors and testing each. 36 = 4 × 9 works because 4 and 9 have no common factor.

The two tests then read different parts of the number. The 4 test looks only at the last two digits, because 100 is already a multiple of 4. The 9 test looks at the digit sum, because 10 leaves remainder 1 on division by 9.

That split is what makes the question solvable: one test fixes the parity of y, the other fixes x + y.

Key facts

• A number is divisible by 36 exactly when it is divisible by both 4 and 9.
• Divisibility by 4 depends only on the last two digits of the number.
• Divisibility by 9 depends only on the digit sum.
• In 5x79856y6 the fixed digits sum to 46, so 46 + x + y must be a multiple of 9.

Common traps

• Splitting 36 as 6 × 6 and applying the 6 test twice — 6 and 6 are not coprime, so that proves nothing.
• Stopping at x + y = 8 and missing the x + y = 17 branch, which is the only one that allows y = 9.
• Finding √25 = 5 and marking a positive value when the question asks for the negative root.

No directly related past PYQ was found.