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

• 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
• 11 years agoHelpfull: Yes(48) No(0)
• 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
• 11 years agoHelpfull: Yes(24) No(0)
• in my solution at the end i did 75/3 .. it should be 75/2=37
  
  SO 37 is the answer
• 11 years agoHelpfull: Yes(4) No(1)
• 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
• 11 years agoHelpfull: Yes(3) No(1)
• 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
• 11 years agoHelpfull: Yes(2) No(3)
• 8 is the right answer.
  
• 11 years agoHelpfull: Yes(2) No(1)
• 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.
• 11 years agoHelpfull: Yes(1) No(0)
• ye kab aya elitmus mein
• 11 years agoHelpfull: Yes(0) No(2)
• 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 ...
• 11 years agoHelpfull: Yes(0) No(0)
• i couldnt understand ur explanation @himnshu upadhay.
  
• 11 years agoHelpfull: Yes(0) No(1)
• 37 is the right answer.
  
• 11 years agoHelpfull: Yes(0) No(0)