Question

Find three smallest consecutive natural numbers such that the difference between one-third of the largest and one-fifth of the smallest is at least 3

Answer 1

Let first least natural number = x
then second number = x + 1
and third number = x + 2

According to the condition `(1)/(3)(x + 2) - (1)/(5)(x) ≥ 3`
5x + 10 - 3x ≥ 45 ...(Multiplying by 15 the L.C.M. of 3 and 5)
2x ≥ 45 - 10 ⇒ 2x ≥ 35

`x ≥ (35)/(12) ⇒ x ≥ 17(1)/(2)`
∵ x is a natural least number
∴ x = 18
∴ First least natural number = 18
Second number = 18 + 1 = 19
and third number h = 18 + 2 = 20
Hence least natural numbers are 18, 19, 20.