Set of 5 Triplets whose Sums and Products are Equal
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Theorem
The following set of $5$ triplets of integers have the property that:
and:
- $\tuple {6, 480, 495}$, $\tuple {11, 160, 810}$, $\tuple {12, 144, 825}$, $\tuple {20, 81, 880}$, $\tuple {33, 48, 900}$
The sum is $981$, and the product is $1 \, 425 \, 600$.
This is the only known such set of $5$ triplets of integers with this property.
Proof
\(\ds 6 + 480 + 495\) | \(=\) | \(\ds 981\) | ||||||||||||
\(\ds 11 + 160 + 810\) | \(=\) | \(\ds 981\) | ||||||||||||
\(\ds 12 + 144 + 825\) | \(=\) | \(\ds 981\) | ||||||||||||
\(\ds 20 + 81 + 880\) | \(=\) | \(\ds 981\) | ||||||||||||
\(\ds 33 + 48 + 900\) | \(=\) | \(\ds 981\) |
\(\ds 6 \times 480 \times 495\) | \(=\) | \(\ds \paren {2 \times 3} \times \paren {2^5 \times 3 \times 5} \times \paren {3^2 \times 5 \times 11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^6 \times 3^4 \times 5^2 \times 11\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 425 \, 600\) |
\(\ds 11 \times 160 \times 810\) | \(=\) | \(\ds 11 \times \paren {2^5 \times 5} \times \paren {2 \times 3^4 \times 5}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^6 \times 3^4 \times 5^2 \times 11\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 425 \, 600\) |
\(\ds 12 \times 144 \times 825\) | \(=\) | \(\ds \paren {2^2 \times 3} \times \paren {2^4 \times 3^2} \times \paren {3 \times 5^2 \times 11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^6 \times 3^4 \times 5^2 \times 11\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 425 \, 600\) |
\(\ds 20 \times 81 \times 880\) | \(=\) | \(\ds \paren {2^2 \times 5} \times 3^4 \times \paren {2^4 \times 5 \times 11}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^6 \times 3^4 \times 5^2 \times 11\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 425, 600\) |
\(\ds 33 \times 48 \times 900\) | \(=\) | \(\ds \paren {3 \times 11} \times \paren {2^4 \times 3} \times \paren {2^2 \times 3^2 \times 5^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^6 \times 3^4 \times 5^2 \times 11\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 425 \, 600\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $981$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $981$