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Re: If k = 10!(1 + 1/2 + 1/3 + 1/4 + 1/5.... [#permalink]
1st) find the prime factorization of 10!

only the primes (2 , 3, 5, 7) will be part of it

Prime 2:
10/2 = 5
——> 5/2 = 2
—————> 2/2 = 1

(2)^8

Prime 3:
10/3 = 3
——-> 3/3 = 1

(3)^4

Prime 5:
10/5 = 2

(5)^2

Prime 7:
10/7 = 1

(7)^1

So
10! = (2)^8 * (3)^4 * (5)^2 * (7)^1

When we multiply 10! through the parenthesis, the effect will be the removal of the prime factors that make up the denominator

Each fraction will not completely remove any of the prime factors EXCEPT (1/7), since there is only ONE Prime 7 that makes up 10!’s prime factorization

This, the other 9 terms will all contain a prime 7 except:

(10!) * (1/7) = NON Multiple of 7

When we add the other 9 terms (a multiple of 7) to the term (10!) * (1/7) (which is a NON multiple of 7 ——-> we get

Multiple of 7 + NON Multiple of 7 = NON multiple of 7

So the result can not be divisible by 7

(D) 7

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Re: If k = 10!(1 + 1/2 + 1/3 + 1/4 + 1/5.... [#permalink]
Moderator:
Math Expert
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