Take a number and create a second number by removing the units digit.
The sum of these two numbers is 1107.
What was your first number?
(In reply to
Answer by K Sengupta)
Obviously the first number cannot have a single digit.
If the first number has 2 digits then the created number will have the maximum magnitude of 99+9=108, which is a 3- digit number. This leads to a contradiction.
Therefore the first number will contain 3 or 4 digits.
CASE 1: The first number has 3 digits
Then, 100a+10b+c+10a+b =1107
=> 11(10a+b) +c=1107
Reducing both sides mod 11, we have:
c==7 (mod 11), giving: c=7
Hence, we must have:
11(10a+b) = 1100
=> 10a +b = 100
Since 0<= a,b<=9, we observe that no solution is possible to the above mentioned equation.
CASE II: The first number has 4 digits
Then, 1000a+100b+10c+d+100a+10b+c =1107
=> 11(100a+10b+c) +d =1107
Reducing this to mod 11, it is easy to show that d=7 by similar arguments as in the previous case.
Hence, we must have:
11(100a+10b+c) =1100
=> 100a+10b+c =100
Substituting a=1 gives b=c=0
For a>=2, 10b+c becomes a negative quantity. This leads to a contradiction.
Accordingly, we must have:
a=1, b=c=0, and d=7
Consequently, the required magnitude of the first number is 1007.
Explanation to Puzzle Answer
