Latest TCS Aptitude Question SOLUTION: what is the remainder when 1234567891011121314151617181 9202122232425262 72829303132333435363738394041424344 is divided by 45?

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(#M40024303) TCS Numerical Ability question Keep an EYE

what is the remainder when 1234567891011121314151617181 9202122232425262 72829303132333435363738394041424344 is divided by 45?

Asked In TCS Dipika Jha (10 years ago)

Unsolved Read Solution (26)

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Read Solution (26) :

Vikash ojha  (10 years ago)

multiples of 45 follow this pattern..
45(sum of digits 9)
90(sum of digits 9)
135(sum of digits 9)
180(sum of digits 9)
225(sum of digits 9)
270(sum of digits 9)
315(sum of digits 9)
360(sum of digits 9)
405(sum of digits 9)
450(sum of digits 9)
495(sum of digits 18)
540(sum of digits 9)
585(sum of digits 18),
so, we saw that the sum of digits are multiples of 9(i.e 9 or 18..)
now, the sum of digits of given number=1+2+3+4+5....+44= n(n+1)/2 = 44*45/2 = 990= multiple 0f 9.
so,this number is a multiple of 45.. so remainder = 0.

Is this solution Helpfull? Yes (34) | No (47)  

Ashish Prakash  (10 years ago)

sum of the digits is 990 so it is divisible by 9 & 45=9*5 so the no is divisible by 9 so we have to divide by 5 only and dividing any no containing 4 at the end will give the output 4 /.... so ans will be 4.

Is this solution Helpfull? Yes (19) | No (14)  

abhishek jaiswal  (10 years ago)

#ans: 9
(concept of divisibility)

Is this solution Helpfull? Yes (13) | No (7)  

Vaibhav Thapliyal  (10 years ago)

The sum of the digits is 990.

Hence it is divisible by 9.

therefore
1234......44/9 = ....6(the last digit will be 6)

lets call this number "a".
Because the number "a" has last digit 6 it will not be divisible by 5.

hence the number 12345....44 is not divisible by 45.

So the challenge is to find a number whose sum is a multiple of 9 and last digit is 5.

To find this number we simply subtract 9 from the given number(12345...44)to get this number...

123456...4335

sum of the digits is still 990 and when it is divided by 9 the last digit of the quotient is 5, which makes it divisible by 5.

Hence the number is divisible by 45.

Therefore we get 9 as the remainder

Is this solution Helpfull? Yes (7) | No (7)  

jaya akula  (10 years ago)

what is the exaxt ans??? some say it is 0..some say 9..others 22...what is the corect ans..??

Is this solution Helpfull? Yes (6) | No (5)  

Arshal Jain  (10 years ago)

suppose we have to check 9821 for divisibility by 9

9+8+1 =18(divisible)
also we could do
9+81 =90(divisible)
or
98+1=99(divisible)
or
18+9=27(divisible)

so in this case simply do n(n+1)/2 i.e. 44 x 45/2 = 990 , which isdivisible by 9 so whole number 1234.......44 is divisible by 9

Is this solution Helpfull? Yes (6) | No (5)  

Padegal Harshitha  (10 years ago)

sum of its digits is 990
the no shld be divisible by 5&9
since it is divisible by 9 i.e 990/9
it shld then be divisible by 5
the last two digits wn subrated by 9 the no is divisible by both 5&9
so ans is 9

Is this solution Helpfull? Yes (4) | No (4)  

Amsaveni Natarajan  (10 years ago)

sum of the num's=n(n+1)/2
=44*45/2
=990
990/45, remainder=0

Is this solution Helpfull? Yes (3) | No (10)  

somesh singh  (8 years ago)

For a number to be divisible by 45, it needs to be divisible by both 9 & 5.

The sum of the digits of the number
n=1234...44 comes out to be 270 which is divisible by 9 but since the number ends in 4, so it isn't divisible by 5.

To make it divisible by 5, we will need to make the last digit 0 or 5.

We can do so by subtracting 4 from the number, but then the number won't remain divisible by 9 as the sum of the digits becomes 270-4=266 which is not divisible by 9.

But if we subtract 9 from the number, the sum of the digits will still be divisible by 9 as the sum of the digits now becomes 270-9=261.
Plus, by subtracting 9 from the number, we have also converted the units place digit to 5 which makes it divisible by 5 also.
So, n-9 is divisible by 45
OR
n mod 45 = 9
which is the answer.

Is this solution Helpfull? Yes (2) | No   

Dipika Jha  (10 years ago)

the exact answer is 9 but i don't know how it came can anyone explain me??

Is this solution Helpfull? Yes (1) | No (4)  

Karthika  (10 years ago)

@ Abhishek..can yu explain?

Is this solution Helpfull? Yes (1) | No (4)  

Arshal Jain  (10 years ago)

as the given number divisible by 9 so we will divide it successively by 3 and 3
so , divide last 4 digit of given number by 3 as in (3 x 3 x 5 = 45) i.e. 4344/3
so resulting 4 digits are 1448 now divide last 2 digit by 3 i.e. 48/3 = 16
so finally the expression becomes ...... xxxxxxxxxxxx16(i.e. 16 as last 2 digits ) and we have 5 in denominator . which on dividing leaves 1 as remainder
now multiply this 1 with 9 so ans is 9

Is this solution Helpfull? Yes (1) | No (2)  

Ajay Poshak  (10 years ago)

when every multiple of 45 has its unit's digit as 0 or 5, then how can be possible that remainder will be 9?

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Ajay Singh Yadav  (9 years ago)

1234567891011121314151617181 9202122232425262 72829303132333435363738394041424344
calculate sum of digit u will get=270 so 2+7+0=9 so remainder is zero

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Prashant Saini  (8 years ago)

sum of digits is 1+2+3......................+44 = 990 hence, number is divisible by 9.
suppose number is A, so we can write
A = 9K .............................. (i)
Now we know that last digit of A is 4, hence last digit of number K should be 6 so that last digit of A would be 4 as can be seen from equation (i)
So last digit of K should be 6. If any number that ends with when divided by 5 leaves remainder 1
so K can be written as
K=5q+1
put this value of K in equation (i) we get
A=9(5q+1)
A=45q+9.
hence remainder is 9.
hope this helps.

Is this solution Helpfull? Yes (1) | No   

shankar kumar  (10 years ago)

ans- 22
(1+2)/3 remainder=0
(1+2+3)/4 rem=2 i.e. 4/2=2
(1+2+3+4)/5 rem=0
(1+2+3+4+5)/6 rem=3 i.e. 6/2=3
same way when divided by odd no rem is 0 nd when divided by even no rem is (number/2)

Is this solution Helpfull? Yes | No (9)  

jaya akula  (10 years ago)

arshal jain...how can u say dat the num is divisible by 9?

Is this solution Helpfull? Yes | No (2)  

Arshal Jain  (10 years ago)

just add the digits and its says that the number is divisible by 9 ....as the sum is divisible by 9

Is this solution Helpfull? Yes | No (1)  

Arshal Jain  (10 years ago)

it was supposed to be 981 not 9821 ...so consider 981

Is this solution Helpfull? Yes | No (1)  

Arshal Jain  (10 years ago)

@vikash ojha ......in what world is a number with unit digit = 5 , can have a multiple with unit digit = 4......?

Is this solution Helpfull? Yes | No (1)  

zainab izhar  (9 years ago)

since the sum is divisible by 45.hence remainder 0
hint:sum=n/2{2a+(n-1)d}

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BS UMA  (9 years ago)

for this type of problems go with relative primes.
45=5*9.as 5 and 9 have 1 as coomon factor, for divisibility by 5 the last no should be divisible by 5 and as per question last digit is 4,4/5=4.....for further reference verify arunsharma book

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Tanmeet Singh Bhalla  (9 years ago)

ans should be zero bcoz sum of all digits is 44(44+1)/2=990
990/45 then remiander 0
the quotient tends to be 22....so all those saying 22 plzz u should practice more knowing the diff about quotient n remiander....n plzz dont confuse others.

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ARUMALLA DHEERAJ  (9 years ago)

add as a number series you get 2*(1+2+...........+9)+1*(10)+2+2*(3+4+.........+9)+2*(10)+2*(11)+1*(12)(4+5+6+7+8)=280
on dividing 280 with 45 we get remainder as 10

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Harsh Dhoot  (8 years ago)

The number is 1234567891011....44

This number is divisible by 9, and when divided by 5 it gives 4 as a remainder. Hence, it could be any of the following 2 forms-

(9a) and (5b + 4) ..... ( a & b are natural numbers)

We may write, 9a = 5b + 4 (since both represent the same number, i.e., 1234567891011....44)

The numbers which satisfy the above equation are 9 (a=1 & b=1), 54 (a=6 & b=10), 99 (a=11 & b=19) and so on. All these numbers are of the form (45k + 9) where k is a non negative integer.

Hence the remainder is 9.

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Deepak kumar singh  (8 years ago)

let no is N.
N/45 means N should be divisible by 5 and 9.
sum of first 44 no= 44(44+1)/2=22*45=990%9=0
so N is divisible by 9
But for N%5==0 unit place should be 0 or 5
so N is not completely divisible by 45
Remainder =4
because when we check N%5 we get remainder it's LAST DIGIT.

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