Let x and y be the largest four-digit and the smallest five-digit numbers, respectively, that when divided by 789 leaves a remainder of 7 in each case. What is the value of |x — y]? 

Given:

Let x and y be the largest four-digit and the smallest five-digit numbers, respectively, that when divided by 789 leave a remainder of 7.

Formula used:

Number = Divisor × Quotient + Remainder

Calculation:

Largest four-digit number: 9999

⇒ Find x such that 9999 = 789 × Quotient + 7

⇒ Quotient = ⌊(9999 - 7) ÷ 789⌋ = ⌊9992 ÷ 789⌋ = 12

⇒ x = 789 × 12 + 7

⇒ x = 9468 + 7 = 9475

Smallest five-digit number: 10000

⇒ Find y such that 10000 = 789 × Quotient + 7

⇒ Quotient = ⌊(10000 - 7) ÷ 789⌋ = ⌊9993 ÷ 789⌋ = 13

⇒ y = 789 × 13 + 7

⇒ y = 10257 + 7 = 10264

⇒ |x - y| = |9475 - 10264|

⇒ |x - y| = 789

∴ The correct answer is option (2).