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# Ordered triples | PRMO 2017 | Question 21

Try this beautiful problem from the Pre-RMO, 2017 based on Sides of Quadrilateral. You may use sequential hints to solve the problem.

Try this beautiful problem from the Pre-RMO, 2017 based on Ordered triples.

## Ordered Triples – PRMO 2017

What is the number of triples (a,b,c) of positive integers such that abc=108?

• is 107
• is 60
• is 840
• cannot be determined from the given information

### Key Concepts

Largest number of triples

Combinatrics

Integers

PRMO, 2017, Question 21

Elementary Number Theory by David Burton

## Try with Hints

First hint

abc=$3^{3}2^{2}$

a=$3^{\alpha_1}2^{\beta_1}$, b=$3^{\alpha_2}2^{\beta_2}$, c=$3^{\alpha_3}2^{\beta_3}$

Second Hint

${\alpha_1}+{\alpha_2}+{\alpha_3}=3$, ${\beta_1}+{\beta_2}+{\beta_3}=2$

Final Step

${5 \choose 2}$, ${4 \choose 2}$

total= ${5 \choose 2} \times {4 \choose 2}$=(10)(6)=60 ways.

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