Q. Find the no. of zeros in 1^1*2^2*3^3*4^4*5^5*6^6*.......*99^99*100^100???

• The number of zeroes in the answer will depend on two things.
  1) on the number of powers of 10. For example 10^10 contains 10 zeroes.
  2) On the number of 5s in the expression.( Because when five gets multiplied with 2 or 4 or any even number then the result will contain a zero in the units place..
  
  SO finding the number of zeroes in powers of 10.
  10^10 contains 10 zeroes.
  10^20 contains 20 zeroes.
  And son on
  Hence the sum would be 10+20+30+40+.100=550.
  But 100^100 can be written as (10*10)^100
  Hence 100 zeroes are also to be counted. Hence number of zeroes from the powers of 10 comes out to be 550+100=650.
  
  Now finding the number of 5s.
  
  5^5 contains 5 fives.
  15^15=(5*3)^15 contains 15 fives..
  .
  And so on..
  Hence total number of fives will be 5+15+25+..+95=500.
  
  But some fives are left uncounted..
  They are..
  25^25=(5*5)^25 and it had 25 fives left uncounted.
  50^50=(5*10)^50 had 50 fives left uncounted.
  75^75=(5*5*3)^75 had 75 fives left uncounted.
  
  Hence the number of fives becomes 500 +15+50+75=650.
  
  
  Hence the number of zeroes becomes 650+650=1300.
  
  
• 11 years agoHelpfull: Yes(4) No(0)
• 2*5 gives 10
  find the exponent of 5
  a no. divisible by 5 will give a factor of 5
  5^5*10^10*15^15*20^20*25^25*...*50^50*...*75^75*...*100^100
  5^5 gives 5^5
  10^10 gives 5^10
  15^15 gives 5^15
  20^20 gives 5^20
  25^25 gives (5^25)*(5*25)
  similar for others
  no. of 5's=5^5*5^10*5^15*5^20*(5^25)^2*...*(5^50)^2*...*(5^75)^2*...*(5^100)^2
  =5^(5+10+15+...+100)* 5^25*5^50*5^75*5^100
  =5^(10*105) *5^250
  =5^1300
  factors of 2 will be much more than factors of 5
  Hence,number of zeroes becomes =1300
  
  
  
• 11 years agoHelpfull: Yes(3) No(0)