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22. What number besides 137 will exactly divide 11371?

23. The quotient being 275, the divisor 383, and the remainder 49, what is the dividend?

24. If the dividend is 2756, the quotient 184, and the remainder 180, what is the divisor?

25. What must 5376 be multiplied by, to make 6521088 ? 26. How many times can 437 be subtracted from 18791? 27. If the sum of 14350 and 7845 is divided by 965, the quotient multiplied by 386, and the product diminished by 761, what will the remainder be?

28. The sum of 250 and 173, being multiplied by their difference, and the product divided by 45, what is the quotient? 29. How many men will it take to do as much work in 1 day, as 368 men can do in 134 days?

30. How many men would be required to do the same work in 16 days?

31. Four men, A, B, C, and D, bought a ship together for $16256; A paid $4756, B paid $763 more than A, and C $256 less than B; how much did D pay?

32. Bought sofas for $9212 and selling them at $67 gained $20 on each; how many were bought?

QUESTIONS.

95. What is Division? 96. The Dividend?

97. Divisor? 98. What

is the answer called? What does it show? 99. Remainder?

100. Make the sign of division. How is it read? 101. How else is division denoted?

102. When a number is divided into two equal parts, what is one of the parts called? 104. When a unit is divided into equal parts, what are the parts called?

106. When the divisor and dividend are like numbers, what is the quotient? When the divisor is an abstract number, what are the dividend and quotient? To what is the product of the divisor and quotient equal?

110. How divide when the dividend has decimals? 113. What is the

general rule? How prove division? 118. How divide by 10, 100, 1000, etc.? When the divisor is greater than 1, with ciphers on the right, how proceed?

119. What is the effect of multiplying the dividend or dividing the divisor? Of dividing the dividend or multiplying the divisor? Of multiplying or dividing both by the same number?

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DEVELOPMENT OF PRINCIPLES.

122. 1. What is the quotient of 24 divided by 6? Ans. 4. 2. Separate the dividend and divisor into factors, and write 2x3x4 them thus, ; what factors are common to both?

2 x 3

3. If you cancel the factor 2, which is common to both, what is the quotient? Ans. 4.

NOTE.-To cancel means to cross out or reject.

4. If you cancel both the 2's and the 3's, what is the effect? Ans. The quotient is not altered. Hence, the following

PRINCIPLES.

123. 1°. Cancelling a factor of a number divides the number by that factor.

2o. Cancelling equal factors of the divisor and dividend does not change the quotient. (Art. 119, 3°.)

124. Cancellation is the method of shortening Division, by rejecting equal factors from the divisor and dividend.

The Sign of Cancellation is an oblique mark drawn across the face of a figure; as, 3, 5, 7, etc.

125. To divide by Cancellation.

5. Divide the product of 14 x 15 x 56 by 8 × 45 × 7.

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EXPLANATION.-Since 8 in the divisor is a factor of 56 in the dividend,

we cancel it in both, retaining 7, the other factor of 56. We also cancel 15, a factor of 45, and 7 a factor of 14, retaining the prime factor 2 in the dividend, and 3 in the divisor; then (7 × 2) ÷ 3 = 43, Ans. Hence, the

RULE.--Cancel all the factors common to the divisor and dividend, and divide the product of those remaining in the dividend by the product of those remaining in the divisor. (Art. 123, 2°.)

NOTE. When a factor cancelled is equal to the number itself, the unit 1 always remains. If the 1 is in the dividend it must be retained; if in the divisor, it may be disregarded.

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15. An agent sold 176 boxes of starch, of 15 lbs. each, at 12 cts.; how many loads of corn, having 9 sacks of 5 bu. each, worth 44 cts. a bushel, will it require to pay for the starch?

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NOTE.-Practical Problems, should first be analyzed, and the operations indicated. Then cancel as before.

16. A farmer bought 9 cows at $25 apiece, and paid for them in hay at $15 a ton; how many tons of hay did it require?

17. How many bags of coffee containing 56 lbs., at 28 cts. a pound, must be given for 8 pieces of muslin, each containing 40 yards, at 8 cts. a yard ?

18. How many barrels of flour worth $8 a barrel, must be given for 45 tons of coal at $6 a ton?

19. A miller bought 7 loads of wheat, each containing 28 bags of 3 bushels each, worth $1.50 a bushel, and paid for it in flour at $7 a barrel; how much flour was required?

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ROPERTIES

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UMBERS.

DEFINITIONS.

126. Numbers are divided into Odd, Even, Prime, and Composite.

127. An Even Number is one that can be exactly divided by 2.

128. An Odd Number is one that cannot be exactly divided by 2; as 3, 5, 7, etc.

129. A Prime Number is one that cannot be exactly divided by any number, except a unit and itself; as 5, 7, 11, etc.

NOTE.-All prime numbers except 2 are odd.

130. Two numbers are Prime to each other when the only number by which both can be exactly divided is a unit or one; as 5 and 6.

131. A Composite Number is the product of two or more factors, each of which is greater than 1; as 21 = 3 × 7.

NOTE.-The least divisor of a Composite Number is a prime number.

132. An Exact Divisor of a number is one which will divide it without a remainder.

One number is said to be divisible by another when there is no remainder.

133. The Factors of a number are the numbers which multiplied together, produce that number. (Art. 80.)

Thus, 7 and 9 are the factors of 63; 3, 4 and 5, of 60.

134. A Prime Factor is a prime number used as a factor. NOTE.-The prime factors of a number are also exact divisors of it.

135. The Reciprocal of a number is 1 divided by that number, Thus, the reciprocal of 4 is 1 ÷ 4, or 4.

Ans. 814

7. Divide 4885970 by 6000.

8. Allowing 200 lbs. to a barrel, how 1 68000 lbs. of beef make?

9. In $1 there are 100 cents; how ma 45650 cents?

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10. How many bales of cotton, each we reach home in

in 36000 lbs. ?

11. If $96000 are divided equally among much will each receive?

12. A pound of cotton has been spun in long, and a pound of wool into a thread many pounds of both together will spin a reach round the world, a distance of 25000 13. If 600 steam engines can do the 496 thousand men, to how many men is 1 119. From the relations of the Divi Quotient, we deduce the following

GENERAL PRINCIPLES OF T

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