Question

If 460 is divided by a natural number, then quotient is 2 more than nine times the divisor and remainder is 5. Find the quotient and divisor.

Answer 1

Suppose divisor = x

∴ Quotient= 9x + 2

Dividend = 460 and remainder = 5

Dividend = Divisor × Quotient + Remainder

∴ 460 = x × (9x + 2) + 5

∴ 460 = 9x² + 2x + 5

∴ 9x² + 2x + 5 – 460 = 0

∴ 9x² + 2x – 455 = 0

∴ 9x² – 63x + 65x – 455 = 0

∴ 9x(x – 7) + 65(x – 7) = 0

∴ (9x + 65)(x – 7) = 0

∴ 9x + 65 = 0 or x – 7 = 0

∴ x = `(-65)/9` or x = 7

However, 460 is divided by a natural number, which cannot be negative.

∴ x = 7

∴Divisor = 7

∴ Quotient = 9 × 7 + 2 = 63 + 2 = 65

∴ Quotient is 65 and divisor is 7.