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5. The multiplier is 1440, the product 10264849920: what is the multiplicand ?

Ans. 6. The product is 6242102428164, the multiplicand 6795634 : what is the multiplier ?

Ans.

EXAMPLES IN DIVISION.

em.

1. Divide 7210473. by 37.

Ans. 24 2. Divide 147735 by 45.

Ans. 3283. 3. Divide 937387 by 54.

Ans.

1 rem. 4. Divide 145260 by 108.

Ans. 1345. 5. Divide 79165238 by 238.

Ans. 6. Divide 62015735 by 7803.

Ans. 7947385 7. Divide 74855092410 by 949998. Ans. 8. Divide 47254149 by 4674. Ans.

:-9 rem. 9. Divide 119184669 by 38473.

Ans. -33788 rem. 10. Divide 280208122081 by 912314.

Ans. —-121 rem. 11. Divide 293839455936 by 8405.

Ans. —-346 rem. 12. Divide 4637064283 by 57606.

Ans. -11707 rem. 13. Divide 352107193214 by 210472.

Ans. —-165534 rem. 14. Divide 558001172606176724 by 2708630425.

Ans.

--24 rem. 15. Divide 1714347149347 by 57143. Rem. 6347. 16. Divide 6754371495671594 by 678957. Rem. 81605. 17. Divide 71900715708 by 57149. Rem. 15785. 18. Divide 571943007145 by 37149. Rem. 12214. 19. Divide 671493471549375 by 47143. Rem. 35411. 20. Divide 121932631112635269 by 987654321. 21. Divide 571943007645 by 37149. Rem. 9691. 22. Divide 171493715947143 by 57007.

CONTRACTIONS IN DIVISION.

CASE 1.

43. When the divisor is a composite number.

RULE.

Divide the dividend by one of the factors of the divisor ; then divide the quotient thus arising by a second factor, and so on, till every factor has been used as a divisor : the last quotient will be the one sought.

EXAMPLES.

OPERATION.

Let it be required to divide 1407'dollars equally among 21 men.

Here the factors of the divisor are 7 and 3. Let the 1407 dollars be first divided equally into 7 piles. 7)1407 Each pile will contain 201 dol

3)201 1st quotient. lars. Let each pile be now di.

67 quotient sought. vided into 3 equal parts. Each one of the equal parts will be 67 dollars, and the whole number of parts will be 21: hence, the true quotient is found by dividing continually by the factors. 2. Divide 18576 by 48=4 x 12,

Ans. 387. 3. Divide 9576 by 72=9x8.

Ans. 133. 4. Divide 19296 by 96=12x8.

Ans. 5. Divide 55728 by 4 x 9X4=144. Ans. 6. Divide 92880 by 2x2x3x2x2. Ans. 1935. 7. Divide 57888 by 4x2x2x2.

Ans. 8. Divide 154368 by 3x2x2.

Ans. 12864. 44. It sometimes happens that there are remainders after division, which are to be treated according to the following

43. What is a composite number? How do you divide when the divisor is a composite number? Repeat the rule.

44. When there are remaiuders, how do you find the true rem mainder? Give the rule.

RULE. 1. The first remainder, if there be one, is a part of the true remainder.

II. Multiply the second remainder, if there be one, by the first divisor, and this product will also form a part of the true remainder.

III. If there be a remainder in any of the after divisions, multiply it by all the preceding divisors, and the final product will be a part of the true remainder : the sum of the several results will be the true remainder sought.

EXAMPLE ILLUSTRATING PRINCIPLES.

1. What is the quotient of 751 grapes, divided by 16?

4)751 Y4=164)187

3
3X4=12

3
15 the true remainder.

Ans. 461

46 ...

DEMONSTRATION OF THE RULE.

In 751 grapes there are 187 sets, (say bunches,) with 4 grapes or units in each bunch, and 3 units over. In the 187 bunches there are 46 piles, 4 bunches in a pile, and 3 bunches over. But there are 4 grapes in each bunch; therefore, the number of grapes in the 3 bunches is equal to 4x3=12, to which add 3, the grapes of the first remainder, and we have the entire remainder 15.

EXAMPLES. 1. Let it be required to divide 4967 by 32.

4)4967 4x8=328)1241 3, 1st remainder. 155 1x4+3=7, the true remainder.

Ans. 1553.. 2. Divide 956789 by 7x8=56.

Ans.

3. Divide 4870029 by 8 X9=72. Ans. 6763944 4. Divide 674201 by 10 x11=110.

Ans. 5. Divide 445767 by 12x12=144. Ans. 3095 874 6. Divide 1913578 by 7 x2x2x2=56.

Ans. 3417136 7. Divide 14610087 by 3x3x2x2x2=72.

Ans. 20291791 8. Divide 2696804 by 5x2x11=110.

Ans. 24516110 9. Divide 936496 by 3 X 4x2x5x6. Ans.

CASE II.

45. When the divisor is 10, 100, 1000, &c.

RULE. I. Cut off from the right hand of the dividend as many figures as there are O's in the divisor.

II. The left-hand figures of the dividend will express the quotient, and the figures cut off the remainder.

EXAMPLE ILLUSTRATING PRINCIPLES."

Divide 3256 by 100.

In this example there are two O's OPERATION. in the divisor, therefore, there are two

100)3256 figures cut off from the right hand of

Ans. 32 the dividend, and the quotient is 32, and 56-100.

5 6

DEMONSTRATION OF THE RULE.

The quotient ought to be 10, 100, 1000, &c., times less than the dividend. But the same figure expresses a number 10, 100, 1000, &c., times greater or less in value, according to its distance from the units' place. By cutting off figures from the right hand, the units'

45. How do you divide when the divisor is 1 with any number of naughts? Give the reason of the rule.

place is removed to the left, and consequently the divi. dend is dirninished 10, 100, 1000, &c., times, according as you cut off 1, 2, 3, &c., figures.

[blocks in formation]

46. When there are ciphers on the right of the divisor.

RULE.

I. Cut off the ciphers by a line, and cut off the same number of figures from the right of the dividend.

II. Divide the remaining figures of the dividend by the significant figures of the divisor, and annex to the remainder, if there be one, the figures cut of from the dividend : this will form the true remainder.

EXAMPLE ILLUSTRATING PRINCIPLES.

OPERATION.

Divide 67389 by 700.

In this example we may re. gard the divisor as a composite 7100)673189 number, of which the factors

96...l remains. are 7 and 100. Hence, we

189 true remain. strike off the 89, and then find that 7 is contained in the remaining figures, 96 times, with a remainder of 1; this we multiply by 100, and then add 89, forming the remainder 189: to the quotient 96 we annex 189 divided by 700 for the entire quotient.

Ans. 96458

46. How do you divide when there are ciphers on the right of the divisor? How do you form the true quotient ?

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