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

## 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

 $\displaystyle 14 \times 50 \times 54$ $=$ $\displaystyle \paren {2 \times 7} \times \paren {2 \times 5^2} \times \paren {2 \times 3^3}$ $\displaystyle$ $=$ $\displaystyle 2^3 \times 3^3 \times 5^2 \times 7$ $\displaystyle$ $=$ $\displaystyle 37 \, 800$

 $\displaystyle 15 \times 40 \times 63$ $=$ $\displaystyle \paren {3 \times 5} \times \paren {2^3 \times 5} \times \paren {3^2 \times 7}$ $\displaystyle$ $=$ $\displaystyle 2^3 \times 3^3 \times 5^2 \times 7$ $\displaystyle$ $=$ $\displaystyle 37 \, 800$

 $\displaystyle 18 \times 30 \times 70$ $=$ $\displaystyle \paren {2 \times 3^2} \times \paren {2 \times 3 \times 5} \times \paren {2 \times 5 \times 7}$ $\displaystyle$ $=$ $\displaystyle 2^3 \times 3^3 \times 5^2 \times 7$ $\displaystyle$ $=$ $\displaystyle 37 \, 800$

 $\displaystyle 21 \times 25 \times 72$ $=$ $\displaystyle \paren {3 \times 7} \times \paren {5^2} \times \paren {2^3 \times 3^2}$ $\displaystyle$ $=$ $\displaystyle 2^3 \times 3^3 \times 5^2 \times 7$ $\displaystyle$ $=$ $\displaystyle 37 \, 800$

## 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.