prob on numbers pls help.....
Q1-there exist a 5 digit numb N with distinct and non zero digits such that it equals the sum of all distinct three digit numbers whose digits are all different and are all digits of N .then the sum of the digits of N is necissarily
a-perfect square b-cube c even d none of these
Q2-find the last non zero digit of 96!
a-2 b-4 c-6 d-8
Q3 -what is the digital sum19^100?pls explain the fundas
a-1 b-4 c-7 d-9
Its definitely an even number
but is it a perfect square/cube ? I think that cant be determined.
Let the 5 digits of the 5-digit number are a, b, c, d and e.
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The last digit of a product of few numbers depends upon how many 2’s, 3’s, 5’s and 7’s are in factors of those numbers.
When 5 is multiplied with any number it results 0 or 5 at the end and when 2 is one of the factor then the last digit is always an even number i.e. either 0, 2, 4, 6 or 8.
The last non-zero digit of any number (>= 2) ends with 2, 4, 6 and 8.
Here in 96! First we ‘ll try to find out number of 2’s , 3’s , 5’s and 7’s
Number of 2’s = 48 + 24 + 12 + 6 + 3 + 1 = 94
Number of 3’s = 32 + 10 + 3 + 1 = 46
Number of 5’s = 19 + 3 = 22
Number of 7’s = 13 + 1= 14.
In the factorial all the 5’s ll result in zeros at the end so we ‘ll have 22 zeros and thus the number of 2’s that makes the last non-zero number is 94 – 22 = 72
Now lets consider the concept of cyclicity
So 2 has a cylicity of 4 and thus 272 would make 6 at the end
So 3 has a cylicity of 4 and thus 346 would make 9 at the end
So 7 has a cylicity of 4 and thus 714 would make 9 at the end.
The last non zero digit should be the last digit of (9 x 9 x 6) = 6
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definition of digit sum is repetitive process of addition of all digits of a number till we get a single digit...
e.g. ds(2442356) = ds(2+4+4+2+3+5+6)= ds(26) = 8
the simplest way to find ds is,find the remainder whn the no. is divided by 9.
e.g. ds(2442356)= R(2442356/9)= 8...
cool isn't it? :-)
working on similar lines...
ds(19^100)= R(19^100/9) = 1
revert me back if u hv any doubt... or reach me at shaheen.csc@gmail.com
regards,
shaheen
hi nishit...
sorry didnt see ur post on digit sum...i ws explaining on similar lines... :-)
regards,
shaheen
hi nishit,
why didnt u consider prime numbers after 7?? e.g. 11,13,17?? wont they affect the rightmost non-zero digit ??
regards,
shaheen
Why number of 5's are subtracted fromnumber of 2's
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The digital sum of any number N is the remainder when N is divided by 9.
Reminder of 19100 divided by 9 = 1
So 1 is the answer
If you dont know what is digital sum then - It is repeated sum of the digits of a number. For example Digital Sum of 235 is 5+3+2=10 ; 1+0 = 1
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