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

Sol.

Le 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.