Hence, we have the following

Rule.

I. Write the multiplier under the multiplicand, placing units of the same order in the same column.

II. Beginning with the units' figure, multiply the multiplicand by each significant figure of the multiplier, and write the first figure of each partial product directly under its multiplier.

III. Then add their partial products, and the sum will be the required product.

Proof.

Write the multiplicand in the place of the multiplier, and find the product, as before. If the two products are the same, the work is supposed to be right.

25. What is the product of the number 728056, multiplied by 50467? X,

26. What is the product of the number 579073, taken 604678 times?

27. What will be the result of taking the number 590587, 79904 times?

28. If the number 9127089 be taken 670456 times, what number will express the result?

29. What is the product, when the number 30726 is multiplied by 97034219?

30. What is the product of the two numbers, 870623 and 91678538?

31. Multiply five thousand nine hundred and sixty-five, by six thousand and nine.

32. Multiply eight hundred and seventy thousand six hundred and fifty-one, by three hundred and seven thousand and four.

33. Multiply four hundred and sixty-two thousand six hundred and nine, by itself.

34. Multiply eight hundred and forty-nine millions six hundred and seven thousand three hundred and six, by nine hundred thousand two hundred and four.

35. Multiply 704 millions 130 thousand 496, by three thousand three hundred and one.

36. Multiply forty-nine millions forty thousand six hundred and ninety-seven, by nine millions forty thousand seven hundred and nine.

Composite Numbers.-Factors.

53. A COMPOSITE NUMBER is one which may be produced by multiplying together two or more numbers.

54. A FACTOR is any one of the numbers which, multiplied together, produce a composite number.

Thus, 2 × 3 = 6, 2 and 3 are the factors of the composite number 6.

53. What is a composite number?-54. What is a factor?

Also, 12 6 x 2 = 3 × 2 × 2, is a composite number, and the factors are 6 and 2; but 6 is a composite number, whose factors are 3 and 2: hence, 3, 2, and 2 are factors of 12.

1. What are the factors of 8? 2. What are the factors of 4?

Of 9?
Of 28?

Of 10? Of 14?
Of 30? Of 32?

55. When the multiplier is a composite number.

1. Multiply 8 by the composite number 6, of which the

ANALYSIS.-If we write 6 horizontal lines, with 8 units in each, the product of 8 × 6 48, will express the number of units in all the lines.

If we divide the horizontal lines into sets of 3 each (as on the left), there will be 2 sets; the number in each set will be 8 x 3: 24, and since there are 2 sets, the whole number of units wiil be 24 x 2 = 48.

If we divide the lines into sets of 2 each (as on the right), there will be 3 sets; the number in each set will be 8 x 2 = = 16, and since there are 3 sets, the whole number of units will be 16 x 3 = 48. Hence,

If the multiplier is a composite number, multiply by the factors, in succession.

Contractions in Multiplication.

56. CONTRACTIONS, in Multiplication, are short methods of finding the product, when the multiplier is a composite number.

55. How do you multiply, when the multiplier is a composite number?

56. What are contractions, in Multiplication?

130

CASE I.

57. When the factors are any numbers.

Rule.

I. Separate the composite number into its factors.

II. Multiply the multiplicand by one factor, and the product by a second factor; and so on, till all the factors have been used: the last product will be the product required.

Examples.

1. Multiply 327 by 12.

The factors of 12 are 2 and 6; they are also 3 and 4, or they are 3, 2, and 2.

58. When the multiplier is 1, with any number of ciphers annexed; as, 10, 100, 1000, &c.

Placing a cipher on the right of a number, is called, annexing it. Annexing one cipher, increases the unit of each place ten times: that is, it changes units into tens, tens into hundreds, hundreds into thousands, &c.; and, therefore, increases the number ten times.

Thus, the number 5 is increased ten times by annexing one cipher, which makes it 50. The annexing of two ciphers increases a number one hundred times; the annexing of three ciphers, a thousand times, &c.: hence, the following

Rule.

Annex to the multiplicand as many ciphers as there are in the multiplier, and the number so formed will be the required product.

57. How do you multiply, when the factors are any numbers ?

Examples.

1. Multiply 254 by 10.
2. Multiply 648 by 100.
3. Multiply 7987 by 1000.
4. Multiply 9840 by 10000.

5. Multiply 3750 by 100.
6. Multiply 6704 by 10000.
7. Multiply 2141 by 100.
8. Multiply 872 by 100000.

CASE III.

59. When there are ciphers on the right hand of one or

both of the factors.

In this case, each number may be regarded as a composite number, of which the significant figures are one factor, and 1, with the requisite number of ciphers annexed, the other. 1. Let it be required to multiply 3200 by 800.

OPERATION.

3200 = 32 × 100;

Then, 3200 × 800

and 800 = 8 × 100.
32 × 100 × 8 × 100,
= 32 × 8 × 100 × 100,
= 2560000.

Hence, we have the following

Rule.

Omit the ciphers, and multiply the significant figures: then place as many ciphers at the right hand of the product as there are in both factors.

58. If you place one cipher on the right of a number, what effect has it on its value? If you place two, what effect has it? If you place three? How much will each increase it? How do you multiply by 10, 100, 1000, &c. ?

59. When there are ciphers on the right hand of one or both the factors, how do you multiply?