‘There’s just not enough money to go round,’ said the president glumly. ‘Look at this great amount being spent on the space program. I need space on my ranch to entertain all these foreign diplomats, not space out there.’

‘No trouble, Mr President,’ said his financial aide. ‘We’ll cut expenditure on the space program to one third of the present amount. That should guarantee you an extra out-house on the ranch.’

The President looked at the figure for space research. The first digit on the left was a 7. ‘What say we remove this 7 and place it on the extreme right,’ he suggested. ‘That looks to give about a third of the total!’

The financial aide was a little concerned about this fanciful line of action, but after consulting his computer, he was amazed to find that the president’s method gave exactly the right answer.

What was the amount allocated for space research?

Hint 1

Write the (unknown) research amount as a string of letters, each letter representing a digit, and build in the given information to obtain an equation.

Hint 2

Don’t give up too soon: space research has a big budget!

Solution

We know that the sum allocated for space research begins with the digit 7, so we write it as 7abcdef ... n , where each letter stands for a digit between 0 and 9. (The n here does not indicate that there are 15 digits overall!)

The other given information is that if we move the 7 to the right hand side, we get exactly a third of the original amount. Thus

7abcdef ... n ÷ 3 = abcdef ... n7.

Now we have to begin the division.

Dividing 7 by 3 gives quotient 2 and remainder 1. Thus from the right hand side, a = 2. Using this on the left, dividing 12 by 3 gives quotient 4 and remainder 0. Thus b = 4. Dividing 4 by 3 gives quotient 1 and remainder 1. Thus c = 1. ...

Continuing in this way for some time until we obtain quotient 7 and remainder 0, we find that the original amount allocated for space research is

$7 241 379 310 344 827 586 206 896 551.

The President will get quite some outhouse!

Extensions

1. We can assume that the first digit is not 7 but some other number. If for example the first digit is 5, we do not get a solution. What happens here?

2. We can assume that the divisor is not 3 but some other number. For example we might leave the first digit as 7 and take 4 as the divisor. Taking 5 as the divisor brings a surprise; what is it?

Hint 1

Hint 2

Solution

Extensions