Answer: Option E
:
E
Use Chinese remainder theorem,
Let the number be N. N is of the form 9A+2 (i.e. N divided by 9 gives a remainder 2)= 10B+3 (i.e. N divided by 10 gives a remainder 3) =11C+5 (i.e. N divided by 11 gives a remainder of 5)
We will first find the number which is of the form 9A+2=10B+3.--------(1)
Here for B=8, A=9, which are the first integer solutions.
Substituting this in equation (1) we get the value 83, the first number which when divided by 9 gives a remainder 2 and which when divided by 10 gives a remainder 3.
The numbers will fall in an AP with the first digit as 83 and common difference 90 (9*10). The general form of the AP is 83+90k when k=0,1,2.....
Now to include the 3${}^{rd}$ condition(i.e divisor 11 and remainder 5) we can write 83+90k=11C+5 -------(2)
Find the first integer value of K which satisfies the equation such that C is also an integer.
Here for K = 5, C = 48, which are the first integer solutions. Substituting this in equation
(2) we get the value 90 (5) + 83= 533, the first number which when divided by 9 gives a remainder 2, which when divided by 10 gives a remainder 3 and which when divided by 11 gives a remainder 5.
Shortcut:- You may be tempted to mark option (a) after a first glimpse at the options.
Since the question asks for the least number, you should be careful while answering.
You can still back calculate using answer option (a)
option (a) 1523 satisfies the condition. Hence 1523-990 = 533 has to be the first such
number. Answer is option (e)
Also note that, the first number has to be $<$ 990.