numbers

show that the product of four consecutive even number is always divisible by 384




Answer:

let the first number be 2x + 2
so, the 4 consecutive numbers are: (2x + 2), (2x + 4), (2x + 6) and (2x + 8)

let F(x) = (2x + 2).(2x + 4).(2x + 6).(2x + 8)
or F(x) = 2.2.2.2.(x + 1).(x + 2).(x + 3)(x + 4)

using Remainder theorem, -1, -2, -3 and -4 are the factors

so, F(x) is divisible by 2.2.2.2.(-1.-2.-3.-4) = 16.1.2.3.4 = 384