### Brain Teasers

# Half Again As Big

What is the smallest integer such that if you rotate the number to the left you get a number that is exactly one and a half times the original number?

(To rotate the number left, take the first digit off the front and append it to the end of the number. 2591 rotated to the left is 5912.)

(To rotate the number left, take the first digit off the front and append it to the end of the number. 2591 rotated to the left is 5912.)

### Hint

The number has 16 digits. I repeat, the number has 16 digits.### Answer

1,176,470,588,235,294x 1.5 =

1,764,705,882,352,941

Here's how you find the number:

You needed to find a repeating fraction such that the number of repeating digits is one less than the number. This would be a prime number, and seven is the lowest number with this property. With this property, every digit in the repeating fraction appears in each place exactly once (i.e., every repeated digit appears as the first digit after the decimal exactly once for n/p where p > n > 0).

Now you need to find any occurrences where the nth digit is followed by the 1.5nth digit. 'n' will obviously be even.

The repeating digits in 1/7 and their corresponding 'n' where the digits are the first digit after the decimal point are:

142857 - repeating digits

132645 - n

The ordered pairs of adjacent values of n are (1,3), (3,2), (2,6), (6,4), (4,5) and (5,1). None of these has the second number equal to one and a half times the first number.

The next prime number p with a repeating fraction for 1/p containing p-1 repeating digits is 17. The repeating digits for 1/17 are:

.0588235294117647

The easiest way to assign 'n' for each digit is to list all of the digit pairs in order:

XX - n (digit)

05 - 1 (1)

11 - 2 (11)

17 - 3 (12)

23 - 4 (5)

29 - 5 (8)

35 - 6 (6)

41 - 7 (10)

47 - 8 (15)

52 - 9 (7)

58 - 10 (2)

64 - 11 (14)

70 - 12 (16)

76 - 13 (13)

82 - 14 (4)

88 - 15 (3)

94 - 16 (9)

Checking the even-numbered values of n in the above table reveals that ALL even values of n less than 2/3 of 17 are followed by the 1.5n digit. The values of n are and their digit places are:

2,3 (11,12)

4,6 (5,6)

6,9 (6,7)

8,12 (15,16)

10,15 (2,3)

So the five smallest numbers with this property are:

1176470588235294

2352941176470588

3529411764705882

4705882352941176

5882352941176470

Notice that 2352941176470588 can be rotated TWICE, yielding 1.5 times the number each time since the 4->6->9 n digits are in order.

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