Latest Elitmus Aptitude Question SOLUTION: Q. Find the maximum value n such that 157! is completely divided by 18^n?

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Q. Find the maximum value n such that 157! is completely divided by 18^n?

Asked In Elitmus Vijay Singh Adhikari (10 years ago)

Unsolved Read Solution (11)

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RAKESH  (10 years ago)

18=2*3^2
no of factors of 3 in 157!=[157/3]+[157/9]+[157/27]+[157/81]=52+17+5+1=75
no of factors of 2 in 157!=[157/2]+[157/4]+[157/8]+[157/16]+[157/32]+[157/64]+[157/128]=78+39+19+9+4+2+1=152
so 157! can be written as a factor 2^152*3^75*5^a*7^b...
now 18^n=2^n*3^2n
for 157! to be completely divided by 18^n,value of 2n should less or equal to 75,
which gives n=37
ans (n)max=37

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Himanshu upadhyay  (10 years ago)

as it is not prime no,so factorising in prime factors
18=3*3*2
so 18 is formed with comb of 2 3's and one 2
so finding no of 3^2 in 157
first calculate no of 3,then divide by 2 as two 3 form 9
157/3+157/9+157/27+157/81(here in each take the greatest intger, leave the decimal part)
52+17+5+1=75,therfore no of 3^2=75/2=37
similarly calculate no of 2's
ie 157/2+157/4+157/8.....
=152,
choosing the lower value i.e 37 as we require both 9 and 2 to form 18
answer=37

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Mirza  (10 years ago)

in my solution at the end i did 75/3 .. it should be 75/2=37

SO 37 is the answer

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swastik  (10 years ago)

ans should be 8
because the series of 157! is like this
1*....*18*....*36*....*54*....*72*....*90*....*108*...*126*...*144*...*157*
so the digit 18,36,54,72,90,108,126,144 would be divisible by 18
so maximum 8, 18 we can afford
so the power of 18 is 8

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Mirza  (10 years ago)

18 = 2x3x3

We have to check either single 2 or double 3 will satisfy this condition.
For 2:
157/2 = 78/2 =39/2 = 19/2 = 9/2 = 4/2 = 2/2 =1
So the highest power of 2 which devide 157 is 78+39+19+9+4+2+1=152

For 3:
157/3 = 52/3 = 17/3 = 5/3 = 1
So the highest value of 3 which divide 157 is 52+17+5+1=75
but there are two 3s so 75/3=25

So the maximum value of n is 25

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nitish kumar  (10 years ago)

8 is the right answer.

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Himanshu upadhyay  (10 years ago)

157.156.155. upto 1
to divide this how many 18 are required.
or to divide this how many 3*3 and 2 are required.
no of three required are 75, therefore no of 9 required will be 37(as a pair of 3 makes 9)
similarly no of 2's will give be 152.
37 9's and 37 2's makes number 18 37 times.rest we don't require.
so answer is 37.

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ravi  (10 years ago)

ye kab aya elitmus mein

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N Arun Singh  (10 years ago)

find number by dividin 157!/3^2 and 157!/2 ...
157/3+157/9+157/27+157/81,...157/2,157/4,157/8+157/16+157/32+157/64+157/128 ...

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pavithra  (10 years ago)

i couldnt understand ur explanation @himnshu upadhay.

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vishal singh thapa  (10 years ago)

37 is the right answer.

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