 ## Arithmetics is the study of numbers, especially the properties of the traditional operations on them.

Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention.

### Primes

Multiplying two numbers is the same as adding the counts of each prime factors, and division is the same as subtracting the counts. To experiment with primes, have a look at Fractran. For example, using numbers made up of the 3 first primes(2, 3, 5), 2250 is equal to `2^1 x 3^2 x 5^3`.

6 * 375 = 2250
6(1,1,0)375(0,1,3)2250(1,2,3)

To find the prime factorization of a number, start by dividing the number by the first prime number 2 and continue dividing by 2 until you get a decimal or remainder. Then divide by 3, 5, 7, etc. until the only numbers left are prime numbers.

### Peasant Multiplication

In the first column, divide the first number by 2, dropping the remainder if any, until 1 is reached. In the second column, write the numbers obtained by successive multiplication by 2. The answer is found by adding the numbers in the doubling column with odd numbers in their first column.

64 x 61
6461
32122
16244
8488
4976
21952
13904+3904
Result: 3904
61 x 64
6164+64
30128
15256+256
7512+512
31024+1024
12048+2048
Result: 3904

When adding 5 to a digit greater than 5, it is easier to first subtract 5 and then add 10.

```7 + 5 = 12.
Also 7 - 5 = 2; 2 + 10 = 12.
```

### Subtraction of 5

When subtracting 5 from a number ending with a a digit smaller than 5, it is easier to first add 5 and then subtract 10.

```23 - 5 = 18.
Also 23 + 5 = 28; 28 - 10 = 18.
```

### Division by 5

Similarly, it's often more convenient instead to multiply first by 2 and then divide by 10.

`1375/5 = 2750/10 = 275.`

### Multiplication by 5

It's often more convenient instead of multiplying by 5 to multiply first by 10 and then divide by 2.

`137×5 = 1370/2 = 685.`

### Division by 5

Similarly, it's often more convenient instead to multiply first by 2 and then divide by 10.

`1375/5 = 2750/10 = 275.`

### Division/multiplication by 4

Replace either with a repeated operation by 2.

```124/4 = 62/2 = 31. Also,
124×4 = 248×2 = 496.
```

### Division/multiplication by 25

```37×25 = 3700/4 = 1850/2 = 925.
```

### Division/multiplication by 8

Replace either with a repeated operation by 2.

```124×8 = 248×4 = 496×2 = 992.
```

### Division/multiplication by 125

```37×125 = 37000/8 = 18500/4 = 9250/2 = 4625.
• a = x – y\$ where `–` means subtraction, a dyadic use of the symbol
• a = -y\$ where `–` means negative, a monadic use of the same symbol