Latest CAT Aptitude Question SOLUTION: Q. Find the no. of zeros in 1^1*2^2*3^3*4^4*5^5*6^6*.......*99^99*100^100???

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

Asked In CAT Vijay Singh Adhikari (10 years ago)

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Diwakar Dahal  (10 years ago)

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.

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RAKESH  (10 years ago)

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

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