## Lesson 1: Integers. Answer to the Tricky Exercise (VI)

October 24, 2011

Isabel was right. Given two numbers, a and b, it is always verified the product

L.C.M(a,b) · H.C.F.(a,b)= a · b

The product of H.C.F and L.C.M. of two numbers is 9072. If one of the numbers is 72, we will find the other applying the result before. If we divide 9072 by 72 we obtain 126.

The question is why?. Because if we want to get L.C.M(a,b) we choose all the factors with the greatest exponent they appear and when we get H.C.F.(a,b) we choose the common factor with the smallest exponent they appear. Multiplying L.C.M(a,b) · H.C.F.(a,b) we find repeated factors but their  product coincides perfectly with a · b where there are the same repeated factors and the factors corresponding to L.C.M(a,b) and  H.C.F.(a,b). Check it with L.C.M(15,45) · H.C.F.(15,45) , for instance

Congratulations Isabel and Lydia! Thanks for your effort Javier. There will be extra points for everybody!

## Lesson 1: Integers. Lyrics and a Tricky Exercise (V)

October 8, 2011

Thanks for your translations. All of them were close to the right translation. There are two points:

Chips, in this context,  are some “little stones or plastic pieces” used to do calculations with integers in an intuitive way. Fried potatoes would not be appropriate.

To mix something up  means confundir con.

Here you are the lyrics and the  rap again

Integers Rap

I-N-T-E-G-E-R

When you know us

you’ll be a star.

I-N-T-E-G-E-R

This is the kind of numbers we are.

Positives, negatives and zero

Positives, negatives and zero

Get to know us

you’ll be a math hero

Use a number line and chips

When you do it so fast

And flips, and flips, and flips.

Multiplying and dividing use the rules

Don’t mix them up

Use the rules.

I-N-T-E-G-E-R

When you know us

you’ll be a star.

I-N-T-E-G-E-R

This is the kind of numbers we are.

Positives, negatives and zero

Positives, negatives and zero

Positives, negatives and zero

Positives, negatives and zero

Get to know us

you’ll be a math hero

Get to know us

you’ll be a math hero

Multiplying and dividing use the rules

Don’t mix them up

Use the rules.

### A tricky exercise

The product of H.C.F and L.C.M. of two numbers is 9072. If one of the numbers is 72, find the other number. Justify your answer.

## Lesson 1: Jokes And Games On integers (IV)

October 1, 2010

### Now time to relax. So, I will tell you a joke.

Several students were asked the following problem:

Prove that all odd integers are prime.

Well, the first student to try to do this was a math student. Hey says “hmmm… Well, 1 is prime, 3 is prime, 5 is prime, and by induction, we have that all the odd integers are prime.”

Of course, there are some jeers from some of his friends. The physics student then said, “I’m not sure of the validity of your proof, but I think I’ll try to prove it by experiment.” He continues, “Well, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is … uh, 9 is an experimental error, 11 is prime, 13 is prime… Well, it seems that you’re right.”

The third student to try it was the engineering student, who responded, “Well, actually, I’m not sure of your answer either. Let’s see… 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is …, 9 is …, well if you approximate, 9 is prime, 11 is prime, 13 is prime… Well, it does seem right.”

Not to be outdone, the computer science student comes along and says “Well, you two sort’ve got the right idea, but you’d end up taking too long doing it. I’ve just whipped up a program to REALLY go and prove it…” He goes over to his terminal and runs his program. Reading the output on the screen he says, “1 is prime, 1 is prime, 1 is prime, 1 is prime….”

Do you understand every single word or expression?

### Here you are a couple of games, practise and have fun at the same time!

Ordering numbers

Integers Jeopardy Game – This game has 4 categories: adding integers, subtracting integers, multiplying integers, and dividing integers. You can play it alone or in teams.

## Lesson 1: Integers III

October 1, 2010

I will not treat some concepts you know perfectly: factor, divisor, divisibility criteria, prime and composite number etc.

However, I provide you some links to revise them at the end of the post.

### 7. Prime factor decomposition

Prime factor decomposition of a number means writing it as a product of prime factors using powers.

This expression is unique for each number. For instance:

240 = 24 · 3 · 5

924 = 22 · 3   7 · 11

### 8. Highest common factor (H.C.F) and lowest common multiple (L.C.M)

In this course, we will find H.C.F and L.C.M of big numbers by using their prime factor decomposition.

### 8.1. Highest common factor (H.C.F)

The Highest Common Factor (H.C.F) of two (or more) numbers is the largest number that divides evenly into both numbers. In other words, the H.C.F is the largest of all the common factors.

FINDING THE H.C.F. OF BIG NUMBERS

For larger numbers you can use the following method:

1. Find prime factor decomposition of all numbers.
2. Find which factors are repeating in all the numbers and choose them with the lowest exponent they appear.
3. Multiply them to get  H.C.F.

Example:

Find the Highest Common Factor (H.C.F.) of 240 and -924.

```240 = 24 · 3 · 5
924 = 22 · 3   7 · 11```

Common factors are 2 and 3 but we have to choose the lowest exponent, this is 22 and 3.

Multiply the factors which repeat in both numbers  to get the H.C.F.

The Highest Common Factor is :  22 · 3 = 12

We write:

H.C.F(240,-924)= H.C.F(240,924)=12

### 8.2. Lowest common multiple (L.C.M)

The Least or Lowest Common Multiple of several integer numbers is the smallest positive number that is a multiple of all those numbers at the same time, except for 0.

The simple method of finding the L.C.M of smaller numbers is to write down the multiples of the larger number until one of them is also a multiple of the smaller number.

FINDING L.C.M. OF BIG NUMBERS

1. Find all the prime factors of both numbers.
2. Find which factors there are in all the numbers  and choose them with the highest exponent they appear.
3. Multiply them to get  L.C.M.

Example:

Find the Lowest Common Multiple (L.C.M.) of 240 and -924.

From the example of finding the H.C.F. we know the prime factors of both numbers.

```240 = 24 · 3 · 5
924 = 22 · 3   7 · 11```

Factors are: 2, 3, 5, 7, 11.

We have to choose them with the highest exponent: 24, 3, 5, 7, 11.

`The L.C.M (240, -924) = L.C.M (240, 924)=  24 · 3   5 · 7 · 11 = 18480`

Practise these concepts as well as word problems on them on the following links

Factors

Divisibility rules

Prime or composite

Prime factorization

Greatest common factor

Least common multiple

GCF and LCM: word problems

## Lesson 1: Integers II

September 28, 2010

### 4. Powers of integers

If  a  is an integer, the product of a  for n  times is denoted by .

an = a · a · …n times….a

is called the base, the repeated factor,

is called the index, the times a is repeated

an  and  itself is called the nth power of  a.

4.1. Sign of the power of an integer

This is a difficult point for you.

If the exponent is a natural number:

• If the base is positive, the power is positive.
• If the base is negative, the power is positive if the exponent is an even number and negative if the exponent is an odd number.

4.2. Properties of Powers

These are the properties of powers. You must learn them by heart.

1.  a0 = 1

2.  a1 = a

3.  an · am = an+m

4.  an:· am =an-m

6. (an)m = an·m

7.  (a : b)n= an: bn

8.  (a · b)n= an·  bn

You can practice in

but the bases are natural numbers, and on the web

On this web you have to click on ABCD. Some of the questions are not avalaible if you don’t pay (IT IS NOT NECESSARY !!!)

### 5. Combined operations

Combined operations are probably the more complicate point on calculations. Remember the order of operations. When you are computing do:

1. What is inside the brackets and square brackets.
2. Powers and roots.
3. Multiplication and division from left to right.
4. Addition and subtraction from left to right.

All types from Difficulty Level 1 Elemental level

All types from Difficulty Level 2 Easy level

All types from Difficulty Level 3 Required level

All types from Difficulty Level 4 Higher level

Finally here you are a video on combined operations, very useful

### 6. Word problems with integers

Word problems with integers require a good understanding of the cocept of integer number as well as a comprehesion of the statement. Read it three times at least.

In this link, you will find not only the solutions but also the explanations.

http://www.mathgoodies.com/lessons/vol5/challenge_vol5.html

These problems have a higher level. They can have hints but no explanation.

## Lesson 1: Integers (I)

September 22, 2010

Vocabulary on Integers

” Lesson 1: Integers” (Notes)

Word Problems on Integers

I Have to Know by the End of this Lesson

### 1. Revising concepts

Last year you learnt the essential notions about integers but you probably need some help to remember. Here you are some links on the main concepts

Integers on number lines

Absolute value and opposite integers

Compare and order integers

Integer inequalities with absolute values

Here you are a funny video “Integers Rap”. Coluld you write a comment giving the translation of the lyrics into Spanish?

### 2. Operations with integers

To add up two integers we do the following:

• If both of them have the same sign, we add up their absolute values and we write their sign
• If they have different sign, we subtract their absolute values and we write by the result the sign of the integer that has the bigger absolute value.

Subtracting a number is adding up its opposite. We add up the minuend to the opposite of the subtrahend.

 Rule Example Two like signs become a positive sign +(+) (+3)+(+2) = 3 + 2 = 5 -(-) (+6)-(-3) = 6 + 3 = 9 Two unlike signs become a negative sign +(-) (+7)+(-2) = 7 – 2 = 5 -(+) (+8)-(+2) = 8 – 2 = 6

Here you are a video on subtracting integers

The following  exercises are elemental. Click on the following links if you want to refresh your knowledge:

All subtraction

MULTIPLICATION AND DIVISION

To work out the product, or quotient, of two integers:

1st Multiply, or divide,  their absolute values

2nd Write + sign if they have like signs or write – sign if they have unlike signs.

Now remember the rule of the signs

When You Multiply …

 Example × two positives you get a positive: (+3 ) · (+2) = +6 × a positive and a negative you get a negative: (-3) · (+ 2) = -6 × a negative and a positive you get a negative: (+3) · (-2) = -6 × two negatives you get a positive: (-3) · (-2) = +6

The same rules are valid for division

 Example : two positives you get a positive: (+12 ) : (+2) = +6 : a positive and a negative you get a negative: (-12) : (+ 2) = -6 : a negative and a positive you get a negative: (+12) : (-2) = -6 : two negatives you get a positive: (-12) : (-2) = +6

Practise multiplication and division on these links

Multiplying signed integers

Dividing signed integers

Practise  these operations with numbers in brackets on the following links: