10. The product of any two or more odd numbers, is an odd number.

11. If an odd number divides an even number, it will also divide the half of it.

12. If an even number be divisible by an odd number, it will also be divisible by double that number.

13. The product of any two numbers is the same, whichever of the two numbers is the multiplier. (Art. 47.) 14. The least divisor of every number is a prime number.

OBS. Hence, in obtaining the least common multiple, the smallest number which will divide any two or more of the given numbers, is always a prime number, and consequently we divide by a prime number. (Art. 102.)

15. Any number expressed by the decimal notation, divided by 9, will leave the same remainder as the sum of its figures or digits divided by 9. The same property belongs to the number 3, and to no other number. Thus, if 236 is divided by 9, the remainder is 2; so, if the sum of its digits, 2+3+6=11, is divided by 9, the remainder is also 2.

Note. Upon this property of the number 9, is based a convenient method of proving multiplication and division.

PROOF OF MULTIPLICATION BY CASTING OUT THE

NINES.

282. First, cast the 9's out of the multiplicand and multiplier; multiply their remainders together, and cast the 9's out of their product, and set down the excess; then cast the 9's out of the answer obtained, and if this excess be the same as that obtained from the multiplier and the multiplicand, the work may be considered right.

Note. To cast out the 9's from a number, begin at the right hand, add the digits together, and, as soon as the sum is 9 or over,

QUEST-What is the least divisor of every number? Obs. In obtaining the least common multiple of two or more numbers, by what kind of a number do we divide? 282. How is multiplication proved by casting out the 9's? Note. Upon what is this method of proof based?

For

drop the 9, and add the remainder to the next digit, and so on. example, to cast the 9's out of 4626357, we proceed thus: 4 and 6 are 10; drop the 9 and add the 1 to the next figure. 1 and 2 are 3, and 6 are 9; drop the 9 as above. 3 and 5 are 8, and 7 are 15; drop the 9, and we have 6 remainder.

Multiply 565 by 356,

Operation.

Proof.

565 The excess of 9's in the multiplicand is 7.

9's "multiplier is

5.

7x5=35; and the excess of 9's is 8.

3390

2825

1695

Prod. 201140. The excess of 9's in the Ans. is also 8.

PROOF OF DIVISION BY CASTING OUT THE NINES.

283. First cast the 9's out of the divisor and quotient, and multiply the remainders together; to the product add the remainder, if any, after division; cast the 9's out of this sum, and set down the excess; finally cast the 9's out of the dividend, and if the excess is the same as that obtained from the divisor and quotient, the work may be considered right.

AXIOMS.

284. In mathematics, there are certain propositions whose truth is so evident at sight, that no process of reasoning can make it plainer. These propositions are called axioms.

An axiom, therefore, is a self-evident proposition.

1. Quantities which are equal to the same quantity, are equal to each other.

2. If the same or equal quantities are added to equal quantities, the sums will be equal.

3. If the same or equal quantities are subtracted from equals, the remainders will be equal.

4. If the same or equal quantities are added to unequals, the sums will be unequal.

QUEST.-283. How is division proved by casting out the 9's?

5. If the same or equal quantities are subtracted from unequals, the remainders will be unequal.

6. If equal quantities are multiplied by the same or equal quantities, the products will be equal.

7. If equal quantities are divided by the same or equal quantities, the quotients will be equal.

8. If a quantity is both multiplied and divided by the same or an equal quantity, its value will not be altered. 9. The whole of a quantity is greater than a part. 10. The whole of a quantity is equal to the sum of all its parts.

OBS. The term quantity signifies any thing which can be multiplied, divided, or measured. Thus, numbers, yards, bushels, weight, time, &c., are called quantities.

285. The following principles will at once be recognized by the pupil as deductions from the four Fundamental Rules of Arithmetic, viz: Addition, Subtraction, Multiplication, and Division.

286. When the sum of two numbers and one of the numbers are given, to find the other number.

From the given sum subtract the given number, and the remainder will be the other number.

Ex. 1. The sum of two numbers is 25, one of which is 10; what is the other number?

Solution. 25-10=15, the other number. (Art. 40.) PROOF. 15+10=25, the given sum. (Art. 184. Äx. 10.)

2. A and B together own 36 cows, 9 of which belong to A: how many does B own?

3. Two farmers bought 300 acres of land together; one of them took 115 acres: how many acres did the other have?

QUEST.-284. What is an axiom? What is the first axiom? The second? Third? Fourth? Fifth? Sixth? Seventh? Eighth? Ninth ? Tenth? Obs. What is meant by quantity? 286. When the sum of two. numbers and one of them are given, how is the other found?

287. When the difference and the greater of two numbers are given, to find the less.

Subtract the difference from the greater, and the remainder will be the less number.

4. The greater of two numbers is 37, and the difference between them is 10: what is the less number?

Solution. 37-10-27, the less number. (Art. 40.) PROOF. 27+10=37, the greater number. (Art. 39. Obs.)

5. A had 48 dollars in his pocket, which was 12 dollars more than B had: how many dollars had B?

6. D had 450 sheep, which was 63 more than E had : how many had E?

289. When the difference and the less of two numbers are given, to find the greater.

Add the difference and the less number together, and the sum will be the greater number. (Art. 39.)

7. The difference between two numbers is 5, and the less number is 15: what is the greater number?

Solution. 15+5=20, the greater number.
PROOF. 20-15=5, the given difference. (Art. 40.)

8. A is 16 years old, and B is 8 years older: how old is B?

9. The number of male inhabitants in a certain town, is 935; and the number of females exceeds the number of males by 115: how many females does the town contain ?

QUEST.-287. When the difference and the greater of two numbers are given, how is the less found? 289. When the difference and the less of two numbers are given, how is the greater found?

290. When the sum and difference of two numbers are given, to find the two numbers.

From the sum subtract the difference, and half the remainder will be the smaller number.

To the smaller number thus found, add the given difference, and the sum will be the larger number.

10. The sum of two numbers is 35, and their difference is 11: what are the numbers?

Solution. 35-11-24; and of 24-12, the smaller number. And 12+11=23, the greater number.

PROOF. 23+12=35, the given sum. (Art. 284. Ax. 10.)

11. The sum of the ages of 2 boys is 25 years, and the difference between them is 5 years: what are their ages?

12. A man bought a chest of tea and a hogshead of molasses for $63; the tea cost $9 more than the molasses: what was the price of each?

291. When the product of two numbers and one of the numbers are given, to find the other number.

Divide the given product by the given number, and the quotient will be the number required. (Art. 74.)

13. The product of two numbers is 84, and one of the numbers is 7: what is the other number?

A's

Solution. 84-7=12, the required number. (Art. 72.) PROOF. 12x7=84, the given product. (Art. 54.)

14. The product of A and B's ages is 120 years, and age is 12 years: how old is B?

15. A certain field contains 160 square rods, and the length of the field is 20 rods: what is its breadth ?

QUEST.-290. When the sum and difference of two numbers are given, how are the numbers found? 291. When the product of two numbers and one of them are given, how is the other found?