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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

Find the number of ordered triples (a, b, c) of positive integers...

Find the number of ordered triples (a,b,c)(a, b, c) of positive integers such that abc=108a b c=108.

Correct answer

60

Step-by-step solution

abc=33⋅22a=3α1⋅2β1,b=3α2⋅2β2,c=3α3⋅2β3α1+α2+α3=3 and β1+β2+β3=35C2 and 4C2 Total =4C2×5C2=10×6=60\begin{aligned} & a b c=3^3 \cdot 2^2 \\ & a=3^{\alpha_1} \cdot 2^{\beta_1}, b=3^{\alpha_2} \cdot 2^{\beta_2}, c=3^{\alpha_3} \cdot 2^{\beta_3} \\ & \alpha_1+\alpha_2+\alpha_3=3 \text { and } \beta_1+\beta_2+\beta_3=3 \\ & { }^5 C_2 \text { and }{ }^4 C_2 \\ & \text { Total }={ }^4 C_2 \times{ }^5 C_2=10 \times 6=60\end{aligned}
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