The first digit of a six-digit number is 1. This digit 1 is now moved from the first digit position to the end, so it becomes the last digit. The new six-digit number is now 3 times larger than the original number. What are the last three digits of the original number?
Answers
1 is the first digit, followed by 5 other digits. Hence the number n can be represented as
n = 100,000 + x ................(1)
When 1 is shifted to the last place, the other digits shift a place each towards the left. In essence, the other five digits together are multiplied by 10, with 1 added as the last digit. And this number is 3 times larger than the original.
3n = 10x + 1 ................ (2)
Substituting from (1)
x = 42857
Hence the number is 142857
Answer:
The last three digits of the original number are either 857 or 858, depending on whether (F/7) is 0 or 1.
Step-by-step explanation:
Let the six-digit number be represented by ABCDEF, where A is the first digit and F is the last digit. We know that A = 1, and we want to find the last three digits of the original number, DEF.
When we move the 1 from the first digit to the last digit, we get the new number BCDEFA. Since this new number is 3 times larger than the original number ABCDEF, we can write:
3(ABCDEF) = BCDEFA
Substituting A = 1, we get:
3(1BCDEF) = BCDEF1
Expanding the left side, we get:
100000B + 10000C + 1000D + 100E + 10F = 3(100000A + 10000B + 1000C + 100D + 10E + F)
Simplifying and collecting like terms, we get:
70000B + 7000C + 700D + 70E + 7F = 299999A
Substituting A = 1, we get:
70000B + 7000C + 700D + 70E + 7F = 299999
We want to find the last three digits of the original number, which are DEF. Notice that DEF appears on the left side of the equation, so we can solve for DEF by isolating it on one side of the equation.
Rearranging the equation, we get:
7(10000B + 1000C + 100D + 10E + F) = 299999 + F
Dividing both sides by 7, we get:
10000B + 1000C + 100D + 10E + F = 42857 + (F/7)
Since F is a digit, (F/7) must be either 0 or 1. If (F/7) = 0, then we get:
10000B + 1000C + 100D + 10E + F = 42857
In this case, the last three digits of the original number, DEF, are simply the last three digits of 42857, which are 857.
If (F/7) = 1, then we get:
10000B + 1000C + 100D + 10E + F = 42858
In this case, the last three digits of the original number, DEF, are simply the last three digits of 42858, which are 858.
Therefore, the last three digits of the original number are either 857 or 858, depending on whether (F/7) is 0 or 1. We can't determine which one it is based on the information given in the problem, so both answers are valid.
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