11762205 = 3 × 4 × 983 × 3 × 735 = 3 × 4 × 2 × 493 × 3 × 5 × 147;
so the G.C.M. is 49 × 3, or 147.
I may here notice there is an ingenious table by Mr. Ellis, published by Philip at 6d., showing graphically the common measures and multiples of numbers up to 36, which makes this matter clear. I give a section of it:—
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ones | · | · | · | · | · | · | · | · | · | · | · | · |
| Twos | · | · | · | · | · | · | ||||||
| Threes | · | · | · | · | ||||||||
| Fours | · | · | · | |||||||||
| Fives | · | · | ||||||||||
| Sixes | · | · | ||||||||||
| Sevens | · | |||||||||||
| Eights | · | |||||||||||
| Nines | · | |||||||||||
| Tens | · | |||||||||||
| Elevens | · | |||||||||||
| Twelves | · |
We find at a glance the primes.
Looking down the line we see the multiples thus, 12 is a multiple of 1, 2, 3, 4, 6. Looking horizontally and moving down, we come to all the measures of each number.
It is also useful for teaching fractions.
Common denominators.We should next proceed to bring fractions to a common denominator preparatory to addition and subtraction. It is not always easy to find a number that will do for all the denominators. We want a common multiple, and of course the smallest we can have is the best. For this we have only to break up the denominators into factors and make up a number which shall contain all these. I would not let the pupils work at first by the mechanical methods sometimes given: 7230 + 346 + 11621.
| Here | - | 230 | = | 2 × 5 × 23 | - | We want therefore as the common denominator 2 × 5 × 23 × 3 × 3 × 3, which is 6210. | ||
| 46 | = | 23 × 2 | ||||||
| 561 | = | 3 × 3 × 3 × 23 |
Addition of fractions.Suppose we want to add 2⁄3 + 3⁄4 - 7⁄8 + 11⁄24. I should write what we may call skeleton fractions below; I mean simply the line; next enter the denominator 24. This is 8 times as large as 3, i.e., we have made the pieces in the first 8 times as small, so we take 8 times as many. Only after working a fair number of sums should children write all in a single fraction thus:—