a tricky question? it depends?

Question

Find a number such that when you move the first digit to the last place,the new number is 1.5 times the original number. (i.e if original number is 1234, the new number is 2341. Note that the answer is not necessarily 4 digits long)

please provide a complete solution! happy figuring!


ps. it only takes a grade 8 math to work this out

Answer 1

I can find several numbers having that property. Here are some:
1176470588235294
2352941176470588
3529411764705882
(In order to check this result, you will need a calculator that can handle at least 16 digit numbers without losing precision.)

These are not the only such numbers. There are many more.

How to find the first number above:
Let x be the number that we seek. The number x must satisfy the following equation:
10 x - 10^n a + a = 3 x / 2
. (n is the number of digits in the number x)
. (a is the first digit of the number x. 0 < a <= 9)

The left side of the equation describes what will happen to the number x when we remove the leftmost digit and move it to the rightmost position.

Now, rearrange the equation to bring all x's to one side.
17 x = 2 ( (10^n) a - a )

Now, let's find some values for n and a that can satisfy the equation. Suppose a = 1. Then we have:
17 x = 2 ( 10^n - 1 )

So, we need a value of n that will make 10^n - 1 be a multiple of 17. The smallest n that works is 16. Let n = 16, and then solve for x. The result is:
1176470588235294

Answer 2

I'm bad at maths, but i am good at programming.

I wrote a program that computes the given question.
were eg. A = 1234 and B would = 2341

And if A*1.5 is equals to B, then print out the numbers.

And i ran the program from range 1 to 99999999 and there are no solutions.

Unless the actual answer is out of my testing range, or my program is actually wrong, then there is no solution other than 0.

Any extra hints?

Answer 3

I can get some near to 1.5x but not exactly. Are there more than one solution? Also any clues? The only thing I can think of is that if for instance the number is abx then bshould be 1.5a but I can't get a number that's exactly 1.5 x.