# Smallest Number which is Sum of 4 Triples with Equal Products

## Contents

## Theorem

The smallest positive integer which is the sum of $4$ distinct ordered triples, each of which has the same product, is $118$:

\(\displaystyle 118\) | \(=\) | \(\displaystyle 14 + 50 + 54\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 15 + 40 + 63\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 18 + 30 + 70\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 21 + 25 + 72\) |

## Proof

\(\displaystyle 14 \times 50 \times 54\) | \(=\) | \(\displaystyle \paren {2 \times 7} \times \paren {2 \times 5^2} \times \paren {2 \times 3^3}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2^3 \times 3^3 \times 5^2 \times 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 37 \, 800\) |

\(\displaystyle 15 \times 40 \times 63\) | \(=\) | \(\displaystyle \paren {3 \times 5} \times \paren {2^3 \times 5} \times \paren {3^2 \times 7}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2^3 \times 3^3 \times 5^2 \times 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 37 \, 800\) |

\(\displaystyle 18 \times 30 \times 70\) | \(=\) | \(\displaystyle \paren {2 \times 3^2} \times \paren {2 \times 3 \times 5} \times \paren {2 \times 5 \times 7}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2^3 \times 3^3 \times 5^2 \times 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 37 \, 800\) |

\(\displaystyle 21 \times 25 \times 72\) | \(=\) | \(\displaystyle \paren {3 \times 7} \times \paren {5^2} \times \paren {2^3 \times 3^2}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2^3 \times 3^3 \times 5^2 \times 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 37 \, 800\) |

## Also see

## Historical Note

Richard K. Guy discusses this result in his *Unsolved Problems in Number Theory* of $1981$, and carries it forward into later editions.

In his *Unsolved Problems in Number Theory, 3rd ed.* of $2004$, the result is presented as:

*It may be of interest to ask for the smallest sums or products with each multiplicity. For example, for $4$ triples, J. G. Mauldon finds the smallest common sum to be $118$ ... and the smallest common product to be $25200$ ...*

However, in the article cited by Richard K. Guy, which appears in *American Mathematical Monthly* for Feb. $1981$, in fact J. G. Mauldon does no such thing.

Instead, he raises the question for $5$ such triples.

David Wells, in his *Curious and Interesting Numbers, 2nd ed.* of $1997$, propagates this, accrediting the result to **Mauldron**, citing that same problem in *American Mathematical Monthly*.

It is also apparent that **Mauldron** is a misprint for **J.G. Mauldon**.

## Sources

- Feb. 1981: J.G. Mauldon:
*Elementary Problems: E2872*(*Amer. Math. Monthly***Vol. 88**,*no. 2*: p. 148) www.jstor.org/stable/2321140

- Sep. 1982: Lorraine L. Foster and Gabriel Robins:
*E2872*(*Amer. Math. Monthly***Vol. 89**,*no. 7*: pp. 499 – 500) www.jstor.org/stable/2321396

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $118$ - 1994: Richard K. Guy:
*Unsolved Problems in Number Theory*(2nd ed.) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $118$ - 2004: Richard K. Guy:
*Unsolved Problems in Number Theory*(3rd ed.): $\text D 16$: Triples with the same sum and same product