Question

The digits of a 3 digit number are in A.P. and their sum is 15.The number obtained by reversing the digits is 792 more than the original number. Find the number.

Answer 1

Hint:Here we assume the numbers following arithmetic progression as\[\left( {x - d} \right)\], \[x\] and \[\left( {x + d} \right)\] by using the first condition we will find the value of x and using the second condition we get the value of d. Finally by substituting the value of x and d we will find the required number.

Complete step-by-step answer:
Here, it is given that the digits of a 3 digit number are in arithmetic progression.
Let us consider the digits of the number as \[\left( {x - d} \right)\], \[x\] and \[\left( {x + d} \right)\], where d is the common difference in arithmetic progression.
Also given that the sum of digits is 15.
Therefore we get \[\left( {x - d} \right) + x + \left( {x + d} \right) = 15\]
On solving we get the value of x,
\[3x = 15\]
Which implies
\[x = 5\]
Now, we have to find the common difference d.
In the question it is also given that the number obtained by reversing the digits is 792 more than the original number
That is Original number = Number obtained by reversing the digits -792
So, \[\left( {x - d} \right)100 + 10x + \left( {x + d} \right) = \left( {x + d} \right)100 + 10x + \left( {x - d} \right) - 792\]
On solving the above equation we get,
\[111x - 99d = 111x + 99d - 792\]
\[198d = 792\]
Let us divide the above equation by 198 on both sides to find d, we get,
\[d = 4\]
By substituting the value of x and d in the digits we get,
The digits are,
\[\left( {x - d} \right) = 5 - 4 = 1\].
\[x = 5\]
\[x + d = 5 + 4 = 9\] .
Hence by arranging the number we get, the given three digit number is 159.
Hence, the original number is 159.

Additional Information:Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value. For example, the series of natural numbers: 1, 2, 3, 4, 5, 6,… is an AP, which has a common difference between two successive terms (say 1 and 2) equal to 1 (2 -1).

Note:Here we can verify the answer using the second condition, we have found the original number is 159.Let us now reverse every digit in the number we get, 951.Let us find the difference between the numbers we get, 951-159=792 which is nothing but the second condition. Hence the answer we have found is correct.