The absolute value

The absolute value of an integer number is the value without taking in account its sign. We will write the absolute value of an integer number b like |b|.

For example, |7|=7 but |-7|=7.

Operations with integers

Addition an subtraction

To add to integer numbers we must:

• If the addends (batugaiak) have the same symbol we add the absolute value and then we will put the symbol plus or minus.
• If the addends have different symbols, we will make the subtraction between the absolute values and then will put the symbol of the biggest addend.

For the subtraction we will operate identically but taking in account the symbol.

For example:

• (+5)+(+9) = +14
• (+5)+(-9) = -4

Multiplication

To multiply two integer numbers we only have to multiply their absolute values and them follow the next rule:

• + · + = +
• + · - = -
• - · + = -
• - · - = +

If the symbols are identical the result will be positive, but if the symbols are not identical the result will bi negative.

For example:

• (+4) · (+5) = +20
• (+3) · (-3) = -9
• (-6) · (+2) = -24
• (-2) · (-6) = 12

Division

In order to make the division we must follow the same process as in the multiplication but.

Operation hierarchy

The order to do operations is the following:

• Parentheses
• Exponents
• Multiply
• Divide
• Add
• Subtract

Divisibility on natural numbers

A number is divisible by an other number when the division between both of them is exact, that means that the rest of the division is zero.

Examples:

• If we divide 36 by 6, the quotient is 6 and the rest 0. In this case we say that 36 is divisible by 6.
• On the other hand, if we divide 33 by 6, the quotient is 5 but the rest is 3. So that, 33 is not divisible by 6.

Divisibility Test (Rules)

We use Divisibility Tests in order to know if a number is divisible by an other number without doing the division.

There are many test or rules:

All the numbers can be decomposed in factor and express them as a product. However, the prime numbers can be decomposed by an only way: a = a·1

Factoring a number is expressing this number as product of prime numbers; it's also known as decomposing in prime factors. For example: the decomposition in prime number of 72 is the next, 72=2·2·2·3·3. We usually write power in order to make easier the writting. So that: 72=2³·3²

Multiples and Lowest Common Multiple (LCM)

A number N is multiple of an other number A if the division between N and A is exact. In other words, if N is divisible by A. For example: 35 is multiple of 5, it will also be of 7.

An easy but hard way of calculating the multiples of a number is adding intself once and again. For example: the multiples of 7 are {7, 14, 21, 28, 35, 42, 49,...} the list of multiples never ends.

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive number that is divisible by all the numbers we are studing.

We will write LCM(a,b) for the Lowest Common Multiple of the numbers a and b.

For example: What is the LCM of 4 and 6?

Multiples of 4 are: {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, ...}

and the multiples of 6 are: {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...}

Common multiples of 4 and 6 are simply the numbers that are in both lists: {12, 24, 36, 48, 60, 72, ....}

So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12=LCM(4,6).

How to calculate the LCM fo two numbers:

Factors and Higest Common Factor (HCF)

One number A is factor of an other number A if the division between N and A is exact. For example: 7 is factor of 35 because 35/7=5.

In order to know how many factors does a number have, there is a methode created by Euler. We decompose the number we are studing in prime factors and we take the powers. After adding 1 to each power we multiply all those numbers and we will have the quantity of factos of the numbers we are studing. For example: 12=2²·3, so the powers are 2 and 1; after adding 1 to each one we have 3 and 2, and the producto of those numbers is 6. So the number of factors of the numbers 12 is six as you can see in the example bellow.

The Higest Common Factor (HCF) of two or more numbers is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.

We will write HCF(a,b) for the Higest Common Factor of the numbers a and b.

For example: What is the HCF of 12 and 20?

The factors of 12 are: {1, 2, 3, 4, 6, 12}

and the factors of 20 are: {1, 2, 4, 5, 10, 20}

Common factors of 12 and 20 are simply the numbers that are in both lists: {1, 2, 4}

So, from this list of the first few common multiples of the numbers 12 and 20, their higest common factor is 4=HCF(12,20).

How to calculate the HCF fo two numbers: