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Olympiad workbook
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Olympiad workbook
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IOQM
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Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

What is the least positive integer by which 2^5 · 3^6 · 4^3 · 5^3 ·...

What is the least positive integer by which 25⋅36⋅43⋅53⋅672^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7 should be multiplied so that, the product is a perfect square?

Correct answer

15

Step-by-step solution

Given number N=25⋅36⋅43⋅53⋅67N=2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7 N=25+6+7⋅36+7⋅53N=218⋅313⋅53\begin{aligned} & N=2^{5+6+7} \cdot 3^{6+7} \cdot 5^3 \\ & N=2^{18} \cdot 3^{13} \cdot 5^3 \end{aligned}  Required number =31⋅51=15\text { Required number }=3^1 \cdot 5^1=15
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