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UPSC Prelims

Mathematics

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Elementary Algebra / प्रारंभिक बीजगणित

Q74 2022, Elementary Algebra

If 15 × 14 × 13 × … × 3 × 2 × 1 = 3^m × n where m and n are positive integers, then what is the maximum value of m?

यदि 15 × 14 × 13 × … × 3 × 2 × 1 = 3^m × n जहाँ m और n धनात्मक पूर्णांक हैं, तो m का महत्तम मान क्या है?

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15 × 14 × 13 × ….. × 3 × 2 × 1 = 15!

We have to find maximum power of 3 in 15!

That will be in 15/3 (= 5) + 5/3 (= 1) = 6 [integer value of 15/3 integer value of 15/9 integer value of 15/27 = 5 + 1 + 0 = 6 ]

Otherwise, in the multiplication of 15 × 14… 2 × 1, 3 is present 15 (one time), 12 (one time), 9 (two times), 6(one time) and 3 (one time). So, total 1 + 1 + 2 + 1 + 1 = 6 times.

Thus, maximum power of 3 will be 6 in 15!

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