Question

The product of four consecutive natural numbers which are multiples of five is 15,000. Find those natural numbers .

Answer 1

Let the four consecutive natural numbers which are multiples of five be x – 5, x, x + 5 and x + 10 respectively.

According to given condition,

∴ (x – 5) (x) (x + 5) (x + 10) = 15000

∴ x ( x + 5 ) (x – 5 )( x + 10 ) = 15000

∴ (x2 + 5x) (x2 + 10x – 5x – 50) = 15000

∴ (x2 + 5x) (x2 + 5x – 50) = 15000

Put, x2 + 5x = m

∴ m ( m – 50 ) = 15000

∴ m2 – 50 m = 15000

∴ m2 – 50m – 15000 = 0

m2 – 150m + 100m – 15000 = 0

∴ m (m – 150) + 100 (m – 150) = 0

∴ (m – 150) ( m + 100) = 0

∴ m – 150 = 0 OR m + 100 = 0

∴ m = 150 OR m = - 100

∵ Natural Numbers can't be negative,

∴ m ≠ - 100 But m = 150

Now re – substituting,

m = x2 + 5x


m = 150
∴ x2 + 5x = 150
∴ x2 + 5x – 150 = 0
∴ x2 +15x – 10x – 150 = 0
∴ x(x + 15 ) – 10 (x + 15) = 0
∴ (x + 15) (x – 10) = 0
∴ x + 15 = 0 OR x – 10 = 0
∴ x = -15 OR x = 10

∵ Natural Number can't be negative
∴ x ≠ - 15 But x = 10

∴ x – 5 = 10 – 5 = 5
∴ x = 10
∴ x + 5 = 10 + 5 = 15
∴ x + 10 = 10 + 10 = 20

∴ The four consecutive natural numbers which are multiples of five are 5, 10, 15 and 20 respectively.