# Powers of natural numbers

## Tips and exercises

It's called a perfect square every natural number can be written as a power of two other natural number.

a perfect square ‹=› a = k2

It's called perfect cube that any natural number can be written as the third power of another natural number.

a perfect cub ‹=› a = k3

Here are some known results in working with power:

1) The last digit of a perfect square is just one of the following

0, 1, 4, 5, 6, 9.

2) Any perfect square is one of the forms 4p or 8q +1.

(Indeed, if n = 2k, then n2 = 4k2 = 4p, and if n = 2k +1, we n2 = 4k(k +1) +1= 8q +1)

3) Any perfect square is one the forms 3p or 3q +1.

(As above, we consider n = 3k or n = 3k +1 or n = 3k + 2 and we raise to square)

4) If a perfect square contains a prime factor decomposition, this factor is actually in power seem to decompose the initial number.

5) The rest of dividing any perfect square through 4 is 0 or 1.

6) A number that ends in one of the figures 2, 3, 7, or 8 is not a perfect square.

7) To show that a number is not a perfect square we can show that he is between two squares of consecutive numbers.

How many natural numbers, integers between two consecutive perfect squares?

Natural numbers are consecutive perfect squares:
02; 12; 22; 32; 42; 52; 62; 72; 82; 92; 102; ...,or, more precise:
0; 1; 4; 9; 16; 25; 36; 49; 64; 81; 100; ...
Between 0 and 1 are 1-0-1=0 natural numbers;
Between 1 and 4 are 4-1-1=2 natural numbers;
Between 4 and 9 are 9-4-1=4 natural numbers;
Between 9 and 16 are 16-9-1=6 natural numbers;
Between 16 and 25 are 25-16-1=8 natural numbers;
Between 25 and 36 are 36-25-1=10 natural numbers;
Between 36 and 49 are 49-36-1=12 natural numbers;

It notes that the results are numbered consecutively. We pick up again what I wrote above except that we write perfect squares as power exponent 2:

Between 02 and 12 are 0 natural numbers, i.e. 2·0;
Between 12 and 22 are 2 natural numbers, i.e. 2·1;
Between 22 and 32 are 4 natural numbers, i.e. 2·2;
Between 32 and 42 are 6 natural numbers, i.e. 2·3;
Between 42 and 52 are 8 natural numbers, i.e. 2·4;
Between 52 and 62 are 10 natural numbers, i.e. 2·5;
Between 62 and 72 are 12 natural numbers, i.e. 2·6;

Generalizing, we can say that between n2 and (n+1)2 exists 2·n natural numbers, where „n” is any natural number.

### Exercises

Exercise 1

Learn two perfect square integers are integers knowing consecutive perfect squares and among them there are 60 integers.

Solve
Let be n2 and (n+1)2 the two consecutive perfect square integers, where
n is a some natural number.

We have the relation: 2·n=60 => n=30.

So the two consecutive integers that are perfect squares 302 and 312.

Keywords: arithmetic, powers, integers, math exercises, examples of power, perfect squares, perfect cube

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