## “Five” – Half a Ten

“Five”

> 5 is a factor of 10, which links to our decimal system (carry one to the next higher place whenever 10 is reached).

All multiples of 5 ends in digit 0 or 5.

> The square of any integer ending in 5 will also have 5 as the last digit (units place).

–> We can even go one step further: for the square of any integer ending in “5”, the last two digits of the results must be 25. For example: 25 × 25 = 625, and 45 × 45 = 2025.

> When a fraction m/n (both m,n are integers) is converted to a decimal, then:

If n = 5 => the decimal has only one digit after the decimal point

If m/n, after converting to decimals, has only one digit after the decimal point, then n = 2 or n = 5

> 5 is a prime number: 5 = 1 × 5

5 = 2^{21} +1 (this type of numbers are called Fermat numbers)

The next Format number is 17 = 2^{22} +1.

> [this item is a bit hard] The numbers in Fermat’s number sequence increases very rapidly: as

2

^{21}+1, 2^{22}+1, 2^{23}+1, 2^{24}+1, … …

We calculate these numbers as:

5, 17, 257, 65537, ..

So the number increase very rapidly. Interesting enough, the first four prime numbers are all prime numbers, yet the fifth one (2^{25} +1) is a composite number, with the smaller divisor being 641.

And once started, it keeps going this way: it has been known that from the fifth Fermat number to the thirty-second Fermat number (which is 2^{232} +1), all of them are composite numbers. It remains open whether there is any prime number in this sequence down the way.

There are lots of facts regarding prime 5: let’s be simple not to mention everything.