last non zero digit
hello
can any one find out the last non zero digit of the following
1. 37!
2. 170^170
thank u
shahid
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can any one find out the last non zero digit of the following
1. 37!
2. 170^170
Ans.
1. the last non zero digit of 37! is 4.
2. the last non zero digit of 170170 is 9.
I have come across this question many times in many places so thought of writing a detail solution.
Please go through it carefully and revert back if you have any problem at any step.
Initially it might seem lengthy but once you have the expertise it wont take more than couple of minutes.
STEP1 Find out the prime factors in the factorial
37! = 2 34 x 5 8 x 3 17 x 7 5 x 11 3 x 13 2 x 17 2 x 19 x 23 x 29 x 31 x 37
STEP2
37! Contains 8 zeros at the end as we have 8 fives in 37!. So we can get the first non-zero digit by dividing 37! With 109 but that is not an easy.
In stead of doing that we ‘ll first divide 37! With 108 and then we ‘ll try to find out the reminder of the new number when divided by 10.
(Try to clearly understand what is the difference between above two)
STEP3
37!/108 = ( 234 x 58 x 317 x 75 x 113 x 132 x 172 x 19 x 23 x 29 x 31 x 37 )/108
= 226 x 317 x 75 x 113 x 132 x 172 x 19 x 23 x 29 x 31 x 37 = N
STEP4
We ‘ll find the reminder of N divided by 10.
226 = 2 x 225 = 2 x 25 = 2 x 2 = 4
317 = 3 x 316 = 3 x 34 = 3 x 1 = 3
75 = 7 x 492 = 7 x (-1)2 = 7 x 1 = 7
113 = 1
132 = 9
172 = 72 = 9
So reminder of N divided by 10 reduces to
4 x 3 x 7 x 1 x 9 x 9 x 19 x 23 x 29 x 31 x 37
Now multiply left to 2 digits get the reminder divided by 10 and then proceed to the right
4 x 3 = 2
2 x 7 = 4
4 x 9 = 6
6 x 9 = 4
4 x 19 (or 9)= 6
6 x 3( or 23) = 8
8 x 9 = 2
2 x 1 = 2
2 x 7 = 4
And that’s the answer 4
for part one it is 4...
hey
HUMRAJ!!!!!!!!
bhai koi assan rasta hai kya.. it takes a bit of time for me...more over for bigger factorials like 3008! or something like that my method fails....
so if u have any shorter method ..plz share....
and for part 2 its 9...
(170)^170 for non zero last digit=> 17^170 mod 10=17^2 mod 10=9
I know its coming.....and I know i can handle it the best....
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One small correction in incognito s method,, His mtd is the right way to solve this sum.. but the last digit of 20! is not 8 as ge told.. it is 4..
Let me explain stepwise
i) Last digit of 10! is 8.. So last digit of 20! is 8*8 = 4.. It follows the power cycle.. For 10! it is 8, 20! it is 4, 30! it is 2 and 40! it is 6.. It keeps on repeating.. I guess everyone is clear with the method :)
All the other steps are exactly same as wat incognito explained!!
i think 1st one is quite easy if u apply ur mind...
you just have to multiply 4*3*2*1=24..thus 4 is the answer
tell u why...lets suppose
u hv started from 37*36*35=now the product has become 0 at the end.isn't it..
likewise when you come to the no. less than 10..
9*8*7*6*5 again the product has come to 0.but if u multiply 4*3*2*1*=it cannt be 0.thus it is the answer...
no need to apply lots of method just apply ur brain....
thank,pleaase comment weather u liked this method or not...
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Hi Humraz,
I think ur answer for 37! is wrong. Answer should me 8. Correct me if im wrong.
My method,
Till 30! last non zero digit will be 2.
After that 2*1*2*3*4*5*6*7 will give last non zero as 8.
Can anyone explain the last non zero digit 30!, 40! and 50! by this method?
FYI, they are 8, 2 and 2 respectively. :)
can any one find out the last non zero digit of the following
1. 37!
2. 170^170
Ans.
1. last non-zero digit of 37! is 4.
2. last non-zero digit of 170170 is 9.