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In the prime factorization of 12! + 15! + 18! + 21! + 24!, what is the exponent of 3?

 Nov 8, 2022
 #1
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The exponent of 3 is 14.

 Nov 8, 2022
 #2
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ok so we break it down 

 

12! has 4 "3" factors and 1 "9" factor

so that's 1+4=5

 

15! has 5 "3's" and 1 "9"

so that's 5+1=6

 

18! has 6 "3's" and 2 "9's"

so that's 6+2=8

 

21! has 7 "3's" and 2 "9's"

so that's 7+2=9

 

24! has 8 "3's" and 2 "9's"

so that's 8+2=10 factors of 3

 

so its basicaly 3^5 + 3^6 + 3^8 + 3^9 + 3^10 

 

simplify that to (3^5)(1 + 3 + 3^3 + 3^4 + 3^5)

 

1 + 3 + 3^3 + 3^4 + 3^5 is not divisible by 3

 

so it's 3^5

 

I think

 Nov 9, 2022

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