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EXAMPLE 1. Multiply 62 by 68.

7 x 6 = 42, the first two figures in the answer ;

2 x 8 = 16, the other two figures in the answer ; therefore 62 times 68 is 4216.

EXAMPLE 2. Multiply 24 by 26.

3 X 2= 6, the first figure in the answer ;
4 x 6 = 24, the other figures in the answer;

therefore 24 times 26 is 624.

EXAMPLE 3. Multiply 69 by 61.

7 x 6 = 42, the first figures in the answer;

9x1= 09, the other figures in the answer ; therefore 69 times 61 is 4209.

ORAL DRILL

Using the above rule find the following indicated products :

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To Multiply by 11. - One or the other of the following cases can be used.

CASE I. - When the number consists of two figures whose sum is less than 10.

Rule.

Write their sum between them.

EXAMPLE 1. 32 x 11 352.
EXAMPLE 2. 54 x 11= 594.

ORAL DRILL

Using the above rule find the following indicated products :

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CASE II. – When the number consists of two figures whose sum is 10 or more.

Rule. - (1) Place the right-hand figure of the number as the right-hand figure of the answer.

(2) Add the figures of the number and place the righthand figure of the sum in the tens' place.

(3) Add the other figure of this sum to the left-hand figure of the number for the left-hand figure or figures of the answer.

EXAMPLE 1. 76 x 11 = 836.
EXAMPLE 2. 87 x 11=957.

ORAL DRILL

Using the above rule find the following indicated products :

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Division is the inverse of multiplication and is the process of finding the number of times one number is contained in another.

In the statement “47 divided by 15 is 345,” 47 is called the dividend, 15 is called the divisor, 3 is called the quotient, and 2 is the remainder.

EXAMPLE

3 quotient divisor 15)47 dividend

45 product of quotient and dividend

2 remainder Multiply the quotient by the divisor and add the reThe result is the dividend.

Check. mainder.

Divisibility of Numbers. — It is an advantage to be able to tell at a glance whether a certain number will be contained in another an exact number of times. The following tests of divisibility are some of the most useful ones.

1. All even numbers are divisible by 2.
2. A number is divisible by 3 if the sum of its digits is divisible by 3.

3. A number is divisible by 4 if the number represented by the right-hand two digits is zero or is divisible by 4.

4. A number is divisible by 5 if it ends with a 5 or a 0.

5. A number is divisible by 6 if it is an even number and the sum of its digits is divisible by 3.

6. A number is divisible by 8 if the number represented by the right-hand three digits is divisible by 8.

7. A number is divisible by 9 if the sum of its digits is divisible by 9.
8. A number is divisible by 10 if it ends with a 0.
9. A number is divisible by 12 if it is divisible by 3 and 4.

10. A number is divisible by 11 if the difference between the sums of the odd-placed and even-placed digits is zero or is divisible by 11.

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Long Division. When all the steps in division are written down the process is called Long Division. Long division is generally used with divisors of two or more places.

Rule. — (1) Write the divisor at the left of the EXAMPLE 1

dividend. 15912 (2) Find how many times the divisor is contained 46)7327 in the smallest group of figures on the left of the 46

dividend, that will contain it, and write the quotient 272 over the right-hand figure of this group. 230

(3) Multiply the divisor by this quotient and place 427 the product under the group or partial dividend used. 414 (4) Subtract this product, and to the remainder 13 annex the next figure of the dividend. Treat the

number so formed as a new partial dividend. EXAMPLE 2 EXAMPLE 3

(5) If any partial dividend will

not contain the divisor, write a zero 30485

in the quotient, then annex the 68) 20675 342)12745

next figure of the dividend, and pro204

1026

ceed as before.
275
2485

(6) Divide as before, and con272

2394

tinue the process until all the figures 3

91

of the dividend have been used. (7) If there is a remainder after the last division, write it with the divisor under it as part of the quotient.

Check. Multiply the quotient by the divisor and add the remainder. The result is the dividend.

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