Latest Elitmus Aptitude Question SOLUTION: for what maximum value of x can (7^n)! be completely divided by 7^x

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for what maximum value of x can (7^n)! be completely divided by 7^x

Asked In Elitmus ashish (9 years ago)

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RAKESH  (9 years ago)

no. of 7's in (7^n)! = 7^n/7 + 7^n/7^2 + 7^n/7^3 + ... + 7^n/7^n

= 7^(n-1)+ 7^(n-2)+ ... + 1
= [7^(n-1)- 1]/ 6

for (7^n)! to be completely divided by 7^x
x(max) should be equal to [7^(n-1)- 1]/ 6

Is this solution Helpfull? Yes (15) | No (3)  

Laxmi   (9 years ago)

7^(n-1)+7^(n-2)+7^(n-3)+...7^0
summation= ((7^n)-1)/6

Is this solution Helpfull? Yes (4) | No (1)  

avinash  (9 years ago)

(7^(n-1))+(7^(n-2))+(7^(n-3))+......upto 7^0

Is this solution Helpfull? Yes (2) | No (1)  

ashish  (9 years ago)

guys please tell me value of x please don't give equations , iss ko to koi bhi nikal le ga....

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Pankaj Yogi  (9 years ago)

the expansion for (7^n)! would look like:
(7^n)! = (7^n).((7^n)-1).((7^n)-2)....((7^n)-r).....3.2.1 ,by defn of factorial
To be divided completely by 7, there are two types of terms in the series:
1. that exhibiting powers of 7 e.g. 7,7^2,7^3,7^(n-1),7^n etc. and,
2. others multiples of 7 e.g. 14,21,28,36,54 etc.
If we try to add up no. of 7's present as a factor in above equation, then depending upon n value we get series as: 1,8,57,400,2801.... as VIVEK shown,for n=1,2,3...where each term is a summation of GP as below: Therefore, nth term is:
term = 7^(n-1) + 7^(n-2) + 7^(n-3) +.....+ 7^(n-r) +.....+ 7^2 + 7^1 + 7^0
which is ultimately the required value of x. Hence, using GP summation formula we get, ((7^n) - (7^0))/(7-1) = ((7^n) -1)/6

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abhishek  (9 years ago)

we say no......

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Vivek singh bhatnagar  (9 years ago)

for n=1
7!/7^x
x=8

n=2
(7^2)!/7^x
x=8

n=2
(7^2)!/7^x
x=8

n=2
(7^2)!/7^x
x=8

n=3
(7^3)!/7^x
x=57

n=4
(7^4)!/7^x
x=400......

depend upon n value if specify....

Is this solution Helpfull? Yes | No (4)  

ashish  (9 years ago)

avinash can u tell me among these options which one is correct
1>
((7^(n-1))-1)/6
2>
n^3
3>
(n+1)^2

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RAHUL KUMAR  (9 years ago)

for x=n , (7^n)! is completely divisible by 7^x.

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Amrita priyadarshi  (9 years ago)

guys i think answer will be n^3..what u say?

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Mani kandan  (9 years ago)

hey guys y confusing me.
pls give the ans with explanaton

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