The child will try in every way to make other combinations and he will try also to divide the prime numbers into factors.
This intelligent and pleasing game makes clear to the child the "divisibility" of numbers. The work that he does in getting these factors by multiplication is really a way of dividing the numbers. For example, he has divided 18 into 2 equal groups, 9 equal groups, 6 equal groups, and 3 equal groups. Previously he has divided 6 into 2 equal groups and then into 3 equal groups. Therefore when it is a question of multiplying the two factors there is no difference in the result whether he multiplies 2 by 3 or 3 by 2; for the inverted order of the factors does not change the product. But in division the object is to arrange the number in equal parts and any modification in this equal distribution of objects changes the character of the grouping. Each separate combination is a different way of dividing the number.
The idea of division is made very clear to the child's mind: 6 ÷ 3 = 2, means that the 6 can be divided into three groups, each of which has two units or objects; and 6 ÷ 2 = 3, means that the 6 also can be divided into but two equal groups, each group made up of three units or objects.
The relations between multiplication and division are very evident since we started with 6 = 3 × 2; 6 = 2 × 3. This brings out the fact that multiplication may be used to prove division; and it prepares the child to understand the practical steps taken in division. Then some day when he has to do an example in long division, he will find no difficulty with the mental calculation required to determine whether the dividend, or a part of it, is divisible by the divisor. This is not the usual preparation for division, though memorizing the multiplication table is indeed used as a preparation for multiplication.
From the above exercises (Table D) others might be derived involving further analysis of the same numbers. For example, one of the possible factor groups for the number 40 is 2 × 20. But 20 = 2 × 10; and 10 = 2 × 5. Bringing together the smaller figures into which the larger numbers have been broken, we get 40 = 2 × 2 × 2 × 5; in other words 40 = 23 × 5.
This is the result for 60:
60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 22 × 3 × 5
For these two numbers we get accordingly the prime factors: 23 × 5; and 22 × 3 × 5. What then have the two larger numbers, 40 and 60 in common? The 22 is included in the 23; the series therefore may be written: 22 × 2 × 5; and 22 × 3 × 5. The common element (the greatest common divisor) is 22 × 5 = 20. The proof consists in dividing 60 and 40 by 20, something which will not be possible for any number higher than 20.
TABLE E