ARITHMETIC. 



Example. Divide 48764312 by 9. 



9)48764312 

 Ans. 5418-568 remainder. 



9 



Proof 48764312 



In this example, we say the 9's in 48, 5 

 times and 3 over; put down 5 and carry 

 3, and say 9's in 37, 4 times and 1 over; 

 put down 4 and carry 1 ; 9's in 16, 1 and 

 7 over; and so on to the end ; there is 8 

 over as a remainder. The proof is ob- 

 tained by multiplying- the quotient by the 

 divisor, and taking in the remainder : 

 this is called '* Short Division," of which 

 we give for practice the following exam- 

 ples. 



1. Divide 4732157 by 



2. 



3. 

 4. 

 5. 

 6. 

 7. 

 8. 

 9. 

 10. 



2 

 3 

 4 

 5 

 6 

 7 

 8 

 9 



11 

 665594765 by 12 



342351742 by 



435234174 by 

 4949:244 by 

 94-947484 by 

 4434983 by 

 994357971 by 

 449^46812 by 

 557779991 by 



The second part of this rule is called 

 " Long Division," for the practice of 

 which we give these directions. 



Count the same number of figures on 

 the left of the dividend as the divisor has 

 in it; try whether the divisor be contain- 

 ed in this number , if not contained there- 

 in, take another dividend figure, and then 

 try how many times the divisor is contain- 

 ed in it. 



To find more easily how many times 

 the divisor is contained in any number; 

 cas* away in your mind all the figures in 

 the divisor, except the left hand one, and 

 cast away the same number from the di- 

 vidend ligures as you did from the divi- 

 sor : the two numbers, being thus made 

 small, will be easily estimated. 



If the product of the divisor with the 

 quotient figure be greater than the num- 

 ber from which it should be taken, the 

 number thought of was too great, the 

 multiplying must be rubbed out, and a 

 less quotient figure used. 



When, after the multiplying and sub- 

 tracting, the remainder is more than the 

 divisor, the quotient figure was 100 small, 

 the work must be rubbed out, and a larg- 

 er number supplied. 



Example. 



Divide 87654213, by 987. 

 987)87654213(88808 Quotient, 

 7896 

 78694 

 7896 

 .7982 

 .7896 

 .8613 

 ^7896 

 .717 remainder. 



88808 

 987 



621663 

 710465 

 799279 

 87654213 proof. 



Ans. 88808|i T. 



To prove the truth of the sum, I mul- 

 tiply the quotient by the divisor, and take 

 in the remainder, which gives the origi 

 nal dividend. 



Examples for Practice. 



1. Divide 721354 



2. 



3. 



4. 



5. 



6. 



7. 



8. 



9. 

 10. 

 11. 

 12. 

 13. 

 14. 



57214372 



67215731 



802594321 



965314162 



43219875 



57397296 



496521 



49446327 



473.24967 



275472734 



43927483 



96543245 



25769782 



by 



by 42 



by 63 



by 84 



by 89 



by 674 



by 714 



by 2798 



by 796 



by 699 



by 497 



by 586 



by 763 



by 469 



A number that divides another without 

 a remainder is said to measure it, and the 

 several numbers that measure another 

 are called its aliquot parts. Thus 3, 6, 9, 

 12, 18, are the aliquot parts of 36. As it 

 is frequently necessary to discover num- 

 bers which measure others, if may be ob- 

 served, 1. That every number ending 

 with an even number, that is, with 2, 4, 

 6, 8, or 0, is measured by 2. 2. Every 

 number, ending with 5 or 0, is measured 

 by 5. 3. Every number, whose figures, 

 when added, amount to an even number 

 of3's or 9's, is measured by 3 or 9 re- 

 spectively. 



