The L. C. M. of 16, 21, 24, 30, 32, may be found as fol

Hence, the L. C. M. = 23 × 3 × 7 × 5 × 4 = 3360.

Since 16 is a measure of 32, it is elided, for any multiple of 32 is also a multiple of 16. The even numbers are divided by 2; the quotients and the odd numbers are written below the horizontal line. This operation is repeated so long as 2 is a measure of more than one number. In the fourth line 3, a measure of 15, is elided. The division by 3 leaves in the fifth line the numbers 7, 5, 4, which are prime to each other.

Therefore, the factors contained in the numbers are 2, 2, 2, 3, and 7, 5, 4.

Hence, the L. C. M. = 2 × 2 × 2 × 3 × 7×5×4 = 3360.

When two or more numbers are prime to each other, their L.C.M. is their product. Thus, the L. C. M. of 3, 5, 7, is

101. If the given numbers are large, and contain no prime factors that can readily be detected, the common factors may be obtained by the process of finding the G. C. M. under like circumstances.

Find the L. C. M. of 1189 and 2117.

Hence, the G. C. M. = 29.

Therefore, 1189 29 x 41; 2117 29 x 73.

= ×

Therefore, the L. C. M.

= =

29 x 41 x 73 1189 × 73.

From this process it will be seen that:

The L. C. M. of two numbers may be found by dividing one of the numbers by their G. C. M., and multiplying the quotient by the other number.

1. A farmer owns 132 acres of wood-land, and 99 acres of pasture; he wishes to divide them into equal lots of the largest possible size. How many lots will there be, and what number of acres in each one? 2. A merchant has 75 yards of one kind of silk, 225 of a second, and 200 of a third; if he cut them into dress patterns of equal size, what is the largest number of yards which each pattern can contain?

3. Simeon Jones has 260 bushels of rye, 384 of oats, and 416 of wheat. He sends his grain to market in bags of equal size. What is the greatest number of bushels which each bag can hold, provided there is no mixture of the different kinds of grain?

4. What width of carpet will fit three rooms, the first 15 feet wide, the second 21 feet, and the third 33 feet?

5. A milk-man has four different measures, holding 2, 3, 5, and 6 quarts respectively. What is the smallest vessel that can be exactly filled by each of them? 6. Find the length of the greatest line that exactly measures the sides of an enclosure 216 yards long and 111 broad.

7. Find the contents of the smallest vessel that may be filled by using a 4-quart, a 5-quart, or a 6-quart

measure.

8. Two apprentices carry 1147 and 961 ivory balls, respectively, from the workshop to the showroom. The balls are carried in baskets of equal size, which are filled and emptied several times. How many balls in a basketful?

9. Find the shortest distance that three lines, 8 feet, 9 feet, and 12 feet long will exactly measure.

CANCELLATION.

102. Cancellation is the operation of striking out equal factors from the dividend and the divisor.

The 3 in the divisor cancels the 3 in the dividend.

Then the fac

tor 2 of the divisor is cancelled, and 2 is cancelled from 12 in the dividend. The resulting dividend is 6 × 1, and the divisor 1 x 1, and therefore the quotient is 6.

Cancellation is used to shorten arithmetical work.

A man carried to a store 49 bushels of potatoes, which he sold at 35 cents a bushel, and took his pay in cents a pound. How many pounds of sugar did SOLUTION. 1 bushel of potatoes is sold for 35 cents.

49 bushels are sold for 49 x 35 cents.
1 pound of sugar is bought for 7 cents.

sugar at 7 he receive?

Number of pounds bought for 49×35 cents is

49 x 35 7