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May 20, 2020

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

Check the Answer


Answer: is 60.

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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