Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
find all the integers which are equal to eleven times the sum of their digits
in my opinion, its not possible for two digit number..
lets say its a two digit number, made of two +ve digits, x & y
10x+y=11(x+y)
so, 10y= -(x)... contradicts!!
3 digit problem has to be solved
100x+10y+z=11(x+y+z)
But it is possible for 3 digit no
If its a two digit number, made of two +ve digits, x & y
10x+y=11(x+y) which aint possible,
THus for three digit no,:
10z + y=89x
for this expression,The values for x,y and z which suits the equation is 1,9,8 respectively
If that number have 1 digit then there are no solution.
If that number have 2 digits then there are no solution.
If that number have more than 4 digits then there are no solution.
Because if that number have digits then :
1000a+100b+10c+d > 11a+11b+11c+11d But , so there are no solution.
Now we just find that number is 198
Hi Vivek,
You should firstly identify the number of digits in the integer.
Lets say we have an integer of the form x1x2x3.....xn (which is an integer of n digits)
So we have 10n-1x1 + 10n-2x2 + 10n-3x3 + ...... 10xn-1 + xn = 11*( x1 + x2 +..... + xn).
You can very easily check that this equation can hold true only for a 3 digit number (ie n = 3) or for n=1 (where the integer will be 0)
In the case of a three digit number, say abc
we have 100a + 10b + c = 11a + 11b + 11c
or 89a = b + 10c
which can hold true only for a = 1 b = 9 and c = 8.
Hence the other integer will be 198.
So there are only two integers " 0 and 198 " for which the above condition will be true.
Hope it helps.
Wish you all the best.
Regards,
Ashwin (IIT MadraS).
First its easy to prove that for , we have
That means the solutions to the above problem is to be found only among numbers with less than 4 digits.
Now, we know that if we denote the sum of digits by S(n), 9 divides n - S(n). Since n = 11 S(n) this means 9 divides S(n) and hence n.
Also 11 divides n and so we have that n is divisible by 99.
Now, its easy to verify that n=198 is the unique solution
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Win Gift vouchers upto Rs 500/-
Register Yourself for a FREE Demo Class by Top IITians & Medical Experts Today !