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Extracting Integer Roots of Numbers |
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Imagine that a stage performer asks a member of the audience to chose a number between zero and a hundred and then compute the fifth power of that number. The fifth power is read to the performer and she instantly tells what the original number was. She has extracted the fifth root of a number that may have five digits and has done it as fast as a computer would have done it.
This sounds like amazingly fast computing but it is very simple. And it can be done with roots other than the fifth. It is just easiest to explain for the fifth roots.
Consider the fifth power of the numbers from 0 to 9.
The Fifth Powers of the First Ten Positive Integers |
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1 | 1 | |
2 | 32 | |
3 | 243 | |
4 | 1024 | |
5 | 3125 | |
6 | 7776 | |
7 | 16807 | |
8 | 32768 | |
9 | 59049 | |
10 | 100000 |
Note that the unit digit of each of these fifth powers is the same as those of the number upon which the power is based.
Suppose for a performance an audience member chooses the number 47. The fifth power of 47 is 229345007. The unit digit of the chosen number must be 7. The performer ignores the first five digits of this number on the right and looks at 2293. The performer who has memorized the approximate values of the fifth powers of the digits knows the value of 2293 is between 4^{5} and 5^{5}. Therefore the chosen number is between 40 and 50 and hence is 47.
For powers other than 5 the unit digit may be different from the base of the power. Here is the values of the unit digit for the first five powers of the digits.
The Unit Digits of the First Five Powers |
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Powers | 2 | 3 | 4 | 5 | |
1 | 1 | 1 | 1 | 1 | |
2 | 4 | 8 | 6 | 2 | |
3 | 9 | 7 | 1 | 3 | |
4 | 6 | 4 | 6 | 4 | |
5 | 5 | 5 | 5 | 5 | |
6 | 6 | 6 | 6 | 6 | |
7 | 9 | 3 | 1 | 7 | |
8 | 4 | 2 | 6 | 8 | |
9 | 1 | 9 | 1 | 9 |
Since the units for the fifth power are the same as those of the first power the cycle starts over again. The cycle has a period of four. Thus the unit digits for the ninth power are the same as those of the first and fifth. Likewise this is true for the powers 13, 17, 21, 25 and so forth.
Note that any power of a number ending in 5 is 5 and of any number ending in 6 is 6. For the third power the cases in which the unit digit of the third power which is not the same as that of the base of the power the sum of digit and the unit digit of the third power is equal to 10. For examples, 2 and 8 and 3 and 7.
In the table there are three cases for which the unit digit of all powers is the same as the base; i.e., 1, 5 and 6. There are two cases in the cycle in digits is two digits long; i.e., 4 and 9. The rest have cycles of period 4.
It would not be too great of a feat of memorization to memorize the above table and thus be able to tell the unit digit of the base from the unit digit of the power. For the K-th power the unit digit of the base number is a function of the remainder when K is divided by 4 (K mod 4). Thus the unit digits for the 23rd powers are the same as those for the 3rd powers.
Shakuntala Devi was a calculating prodigy from India. She could do amazing computations in her head. She was tested by Arthur Jensen, a professor of psychology at the University of California at Berkeley. He asked her to give the seventh root of 170,859,375 and was amazed that she gave him the correct answer of 15 before he could finish writing down the number in his notebook. The unit digit being 5 told her that the unit digit of the root was 5. Eliminating the seven digits on the right gave her 17. This number is between 1 and 128=2^{7} so the root has to be 15. Jensen gave her other challenges which she successfully accomplished but finding the two digit number whose seventh power is know is something that can be done quickly if one has memorized the seventh powers of the single digits. Discarding the first seven digits on the right is a matter of looking at the numbers that are greater than 10^{7}.
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