The digits of a three digit number are in AP&their sum is 15.The number obtained by reversing the digits is 594 less than the original number.Find the number?
the ap = a, a+d, a+2d, so
3a+3d = 15
a+d=5
so, the number is 159 or 258 or 357
reversed, it is either 951 or 852 or 753
Looks like original is 852 - 258 = 594.
To solve this problem, let's break it down step by step.
Step 1: Represent the three-digit number.
Let the three-digit number be represented as 100a + 10b + c, where a, b, and c are the digits of the number, and 'a' is the hundreds place digit, 'b' is the tens place digit, and 'c' is the units place digit.
Step 2: Formulate the given information into equations.
We are given two pieces of information:
1. The sum of the digits is 15: a + b + c = 15
2. The number obtained by reversing the digits is 594 less than the original number: 100c + 10b + a = (100a + 10b + c) - 594
Step 3: Simplify the equations.
The second equation can be simplified by combining like terms:
99c - 99a = -594
Divide both sides by 99 to simplify further:
c - a = -6 [Equation 1]
Step 4: Solve the system of equations.
Now, we have two equations:
a + b + c = 15 [Equation 2]
c - a = -6 [Equation 1]
We can eliminate variable 'c' by adding Equation 1 and Equation 2:
(a + b + c) + (c - a) = 15 - 6
b + 2c = 9 [Equation 3]
Step 5: Find the possible values of a, b, and c.
To find the values of 'a', 'b', and 'c' that satisfy Equation 3, we can substitute different values for 'b' and calculate the corresponding value of 'c'. Since the sum of 'a', 'b', and 'c' is 15, we can try different values that add up to 15.
For example, if we let 'b' be 3, then 'a' + 'c' = 12 (15 - 3). In this case, 'a' can be 6, and 'c' can be 6 as well, which satisfies Equation 1. Therefore, one possible solution is:
a = 6
b = 3
c = 6
Step 6: Calculate the three-digit number.
To find the three-digit number, substitute the values of 'a', 'b', and 'c' into the representation: 100a + 10b + c.
Using the values above, we get:
Number = 100(6) + 10(3) + 6
Number = 600 + 30 + 6
Number = 636
Therefore, the three-digit number that satisfies the given conditions is 636.