Question

the product of four consecutive positive integers is 1680, find numbers

Answer 1

Product of the 4 consecutive number is 1680

1st Step: Let's get an estimate of the number:

Let the average of the 4 numbers be x

x⁴ = 1680

x = 6.4

⇒ These are 4 consecutive numbers, therefore the average number is one that is between the 2nd and the 3rd number.

2nd Step: Find the 4 numbers:

⇒ These are 4 consecutive numbers, therefore the average number is one that is between the 2nd and the 3rd number.

⇒ It is a reasonable guess to say that the 2nd number is 6 and the third number is 7. Therefore the first number is 5 and the last number is 8.

Guess: The numbers are 5,6,7 and 8

Check: 5 x 6 x 7 x 8 = 1680 (Verified)

Answer: The numbers are 5, 6, 7 and 8

Answer 2

Let the four consecutive integers be x, x+1, x+2, x+3

Now (x)(x + 1)(x + 2)(x + 3) = 1680

Multiply each one of them we get

(x^2 + x)(x + 2)

(x^3 + x^2 + 2x^2 + 2x)(x + 3)

x^4 + 6x^3 + 11x^2 + 6x = 1680

By assuming numbers we can try   5

So 5^4 + 6(5^3) + 11(5^2) + 6(5) = 1680

So the numbers are 5, 6, 7 and 8