Sunday, November 30, 2014

Division and Factors

A multiplication puzzle like this


6x = 12

Can be thought of as a puzzle where you are given a number 12, a factor of Twelve, 6, and you have to find the paired factor x.

For simple division it is enough to know your times tables to see that x =2.

However the puzzle below is not so straight forward as we are unlikely to know our Eighteen times table.


18x = 54

A knowledge of factors and multiples can help.
First let us examine the simpler 6x = 12 by sharing apples again.

Twelve apples shared between six boxes.

This problem can be shared between three people each carrying four apples and two boxes.

Each person has four apples to share among two boxes.
This is possible because the number of apples 12, and the number of boxes 6 have the factor 3 in common.

The result is two apples in each box .

Let us return to the problem 18x = 54 by sharing out apples.

Here we have fifty four apples to distribute over eighteen boxes.

We might not know the Eighteen times table but could know that Fifty Four is in the Six times table and the Nine times table and also that Eighteen is in the Six times table and the Nines times table.

Six and Nine are common factors of Fifty Four and Eighteen.

Since Nine is larger than Six we work with Nine as the common factor.

9 x 6 =54 and 9 x 2 = 18

Get nine people to each carry six apples and two boxes.

Each person shares six apples over the two boxes giving three apples in a box

So Fifty Four divided by Eighteen is Three, 54 ÷ 18 = 3.

The Game of Maths

We start with a game with eighteen boards and play fifty four counters shared over the eighteen boards.

This produces the same result as

Then instead of one player playing Fifty Four over eighteen boards allow nine players to split the game between them so now each person is playing Six over two boards.

which gives the result

An important point to note is that the hand can be obtained from by removing the Nine holders for the counter cards and the board cards. This is called cancelling the multiplication.

We record cancelling


Saturday, November 29, 2014

Common Factors

Consider when there are lots of people and each person has two apples and one box.

One person One times two apples and one box

Two people Two times two apples and two times one box.
Two is a factor of Four and Two

Three people
Three times two apples and three times one box.
Three is a factor of Six and Three.

Four people

Four times two apples and four times one box,.

Four is a factor of Eight and Four.

Each person adds two apples and one box .

The number of apples will be the number of people times Two and the number of boxes will be the number of people times One.

The number of people will be a factor of the number of apples and the number of boxes.

Another example.

Each person has five apples and three boxes .

Two people

2 x 5 apples, 2 x 3 boxes, 2 is a factor of 10 and 6, 6 and 10 are in the 2 times table

Six people

6 x 5 apples, 6 x 3 boxes, 6 is a factor of 30 and 18, 18 and 30 are in the 6 times table.

Common factors occur when different numbers of objects are multiplied the same number of times.

When two numbers appear in the same times tables then they have a common factor.

Factors are an important factor in simplifying division.

Factors and Multiples

All numbers appear in the Ones times table.

All numbers appear only in their own times table and the Ones times table.

Some numbers appear in more times tables than their own and the Ones times table.

For example consider the numbers Seven and Twelve.

Take seven apples and put them in boxes. We can put all seven apples in one box or one apple in seven boxes.


Now take twelve apples and put them in boxes. We can;

Put all twelve apples on one box

Put six apples in two boxes

Put four apples in three boxes

Put three apples in four boxes

Put two apples in six boxes

Put one apple in twelve boxes

In the game of maths Twelve can be arranged in a variety of equal rows whereas there is only one row arrangement for Seven, ignoring any rotations of the board.

Besides one row of counters Twelve can also be arranged as or whereas Seven can only be arranged in rows as or in columns as on rotation .

A number with only one row arrangement is called a Prime number.

  1. One is not a prime number because
  2. One is what is being counted and
         rotated is the same as

More on prime numbers later.

Card hands that give Twelve

Twelve is in the multiplication tables of One, Two, Three, Four, Six and Twelve. We say Twelve is a multiple of One, Two, Three, Four, Six and Twelve.

One, Two, Three, Four, Six and Twelve are called factors of Twelve.

We sometimes say Three times Four makes Twelve, things are made in a factory from the Latin factor meaning doer or maker and Three and Four are together key factors (doers) in making Twelve.

The cards above show that if you know one factor of a number then there is another factor, its soul mate as it were, which multiplied by the first factor makes the number. Factors always occur in pairs.

For Seven the only pair is One and Seven.

For Twelve the pairs are: One and Twelve; Two and Six; Three and Four.


                 4 and 3 are factors of 12                            12 is  a multiple of 4 and 3