Math CentralQuandaries & Queries


Question from Nabila:

The product of four consecutive positive integers in 24024
What is their sum?

Hi Nabila,

An algebraic approach would be to let $n$ be the smallest of the four consecutive positive numbers then the four consecutive numbers are n, n+1, n+2 and n+3. I know something about the sum and product of the roots of a polynomial so I might consider the polynomial

\[[x-n][x - (n+1)][x - (n+2)][x - (n+3)].\]

You can then expand this to a quartic of the form $x^4 + ax^3 + bx^2 + cx + d$ and use facts about the sum of the rots of a polynomial from Math is Fun to to find the sum of the roots.

I prefer a different approach. The four consecutive positive integers are almost the same and if they were all $k$ then $k^4$ would be 24024. Hence to solve $k^4 = 24024$ I used my calculator and found that $24024^{0.25} = 12.45.$ Hence I think one of the four consecutive positive integers is 12. The four numbers I then found by experimenting.

I hope this helps,

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