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# Ordered triples | PRMO 2017 | Question 21

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

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

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

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

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

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

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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

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

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

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

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

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

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