The last digti of a six-digit number it 2. If the 2 is moved to the start of the number, the new six-digit number is only a third of the original number. Find the original number.
2 A B C D E
X 3
= A B C D E 2
NOTE: PLS DON'T CONFUSE X WITH x. IT IS A MULTIPLCATION SIGN
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To solve this problem, we need to set up a mathematical equation based on the given information. Let's break down the problem step by step.
Step 1: Define the variables
Let's represent the original six-digit number as ABCDE2, where A, B, C, D, and E are digits between 0 and 9.
Step 2: Formulate the equation
According to the problem, if we move the digit 2 to the start of the number, the new six-digit number becomes A, B, C, D, E, 2. It is also mentioned that this new number is one-third of the original number.
Therefore, we can set up the equation: ABCDE2 = (1/3) × A, B, C, D, E, 2.
Step 3: Simplify the equation
To simplify the equation, we can eliminate the fraction by multiplying both sides of the equation by 3:
3 × ABCDE2 = A, B, C, D, E, 2.
This gives us: 300000A + 30000B + 3000C + 300D + 30E + 6 = 100000A + 10000B + 1000C + 100D + 10E + 2.
Step 4: Solve for the unknown variables
Now, let's collect like terms and solve for the variables. Subtracting 100000A from both sides, 10000B from both sides, 1000C from both sides, 100D from both sides, and 10E from both sides, we get:
299000A + 20000B + 2000C + 200D + 20E + 6 = 2.
Moving all the variables to one side, we have:
298998A + 19998B + 1998C + 198D + 19E = -4.
Step 5: Find the possible solutions
Since we are dealing with whole numbers, we know that the left side of the equation must be divisible by 19 in order for the equation to have a solution. Through trial and error, we can find that when A = 4, B = 7, C = 2, D = 1, and E = 3, the left side of the equation equals -4.
Therefore, the original number is 472132.
To summarize:
The original six-digit number is 472132.