If the 9-digit number 94x257y36 is exactly divisible by 88, what is the maximum possible v… — State PSC Exam

If the 9-digit number $94x257y36$ is exactly divisible by $88$ , what is the maximum possible value of the mathematical expression $(5x - 3y)$ ?

Cevap: 27

By applying the divisibility rules of 8 and 11, we isolate a finite set of possible digit combinations for x and y. Evaluating

(5x - 3y)

across all mathematically valid pairs proves that the pair

x=6, y=1

produces the highest numerical output of 27.

Determine the co-prime factors of the divisor.

The number must be divisible by 8 and 11, since 88 = 8 * 11 and GCD(8, 11) = 1.

Applying individual divisibility rules for co-prime factors is the standard method for composite divisors.

Apply the divisibility rule for 8 to the last three digits (

y36

The possible values for

y

are 1, 3, 5, 7, and 9.

A number is divisible by 8 if its last three digits form a multiple of 8. 136, 336, 536, 736, and 936 all yield a remainder of 0 when divided by 8.

Apply the divisibility rule for 11 to the entire number.

The expression

(x + y + 4)

must be a multiple of 11.

The rule for 11 requires the difference between the alternating sum of digits to be divisible by 11. Odd positions sum to

(20 + x + y)

and even positions sum to

16

. Difference =

(x + y + 4)

Find all valid

(x, y)

coordinate pairs.

The valid digit pairs are (6, 1), (4, 3), (2, 5), (0, 7), and (9, 9).

Since

x

and

y

are single digits between 0 and 9,

(x + y + 4)

can only equal 11 or 22. This implies

x + y = 7

x + y = 18

Evaluate the target expression

(5x - 3y)

for all valid pairs.

The values obtained are 27, 11, -5, -21, and 18 respectively. The absolute maximum is 27.

Testing all valid edge cases guarantees that the maximum possible value is correctly identified without assumptions.

Solving for unknown digits using combined co-prime divisibility rules and optimizing a linear algebraic expression based on finite digit constraints.

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