2022_IAS_Qn A74. If 15×14×13×……..×3×2×1=3^m × nWhere m and n are positive integers,

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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 ?
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If you want to find the greatest power of 3 in the product of the first 15 natural numbers, first divide by three. You get 15/3 = 5. But there are two threes in 9, so to account for that, divide 15 by 9. You get 1.67. Consider the greatest integer in this value, which is equal to [1.67] = 1. The greatest power of 3 in the product 1*2*3*....*14*15 = We stop with 9 because it is the largest power of three less than or equal to 15.

Consider another example: What is the greatest power of 2 in the product of the first 20 natural numbers
The greatest power of 2 is [20/2]+[20/4]+[20/8]+[20/16] = [10]+[5]+[2]+[1] = 10+5+2+1 = 18, where the square bracket indicates the greatest integer function. Here, there are four twos in 16. We consider the first two while dividing 20 by 2, the second while dividing by 4, the third while dividing by 8, and finally, the fourth while dividing by 16. We stop with 16 because it is the largest power of two less than or equal to 20.

Now consider a very simple example: What is the greatest power of 2 in the product of the first 5 natural numbers?
The greatest power of 2 is [5/2]+[5/4]=[2]+[1] = 2+1=3.
Alternatively, 1*2*3*4*5 = 1*(2)*3*(2*2)*5, which gives the greatest power of 2 as 3.

AbdusCSATMATH
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Why we divided it by 3 and 9??? Help pls sir

tanishachaurakoti
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sir one se lekar 18 tak 3 kitne times ayega is concept pe please video banaye Q1) ? HOW MANY TIMES WILL 3 COME ?? I FOUND ANSWER AS 7 TIMES... please guide and confirm

vaishakhsonawane
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Sir if thwy will ask value of n then what it would be??

srivalli
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SIR ABHI BHI NAHI SAMJA PEHLE PLEASE QUESTION EXPLAIN KARTE HO KYA PLEASE 3 *3 i.e. 3 square is 9 toh fir ap 9 ko konse value se multiply karoge toh fir answer 15 ayega...? 3*3=9 i.e. M then what should be value of N to bring 15 as answer

vaishakhsonawane
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Sir abhi clear nhi h koi aur trick plzzz

sanjanamaurya