108. 1. What is the product of 4 times 5? What are 4 and 5 of their product?

2. What is 4 of 16? Of 24? What is 7 of 14? Of 28? 3. What numbers will exactly divide 18? 24? 36? 72? 4. Give the exact divisors of 42. 96. 108. 48. 32?. 5. What are the factors of 30? 24? 40? 56? 64? 6. What numbers between 0 and 10 can not be divided by any number except themselves and 1? Between 10 and 20? 7. What numbers between 0 and 10 can be divided by other numbers than themselves and 1? Between 10 and 20?

DEFINITIONS.

109. An Integer or Integral Number is one that expresses whole units.

Thus, 281, 36 houses, 46 men, are integral numbers.

110. An Exact Divisor of a number is an integer that will divide it without a remainder.

Thus, 2, 4, 6 and 12 are exact divisors of 24.

111. The Factors of a number are the integers which being multiplied together will produce the number.

Thus, 6 and 8 are factors of 48.

112. A Prime Number is one that has no exact divisors except itself and 1.

Thus, 1, 3, 5 and 7 are prime numbers.

113. A Composite Number is one that has exact divisors besides itself and 1.

Thus, 18 and 24 are composite numbers, for 18 is divisible by 6, and 24 by 8.

114. An Even Number is one that is exactly divisible by 2.

Thus, 2, 4, 6, 8, etc., are even numbers.

115. An Odd Number is one that is not exactly divisible by 2.

Thus, 1, 3, 5, 7, 9, etc., are odd numbers.

DIVISIBILITY OF NUMBERS.

116. In determining by inspection the divisibility of numbers, the following, facts will be found valuable.

1. Two is an exact divisor of any even number.

Thus, 2 is an exact divisor of 12, 16, 30 and 44.

2. Three is an exact divisor of any number, the sum of whose digits is divisible by 3.

Thus, 3 is an exact divisor of 312, 135, 423, and 3816.

3. Four is an exact divisor of a number, if the number expressed by its two right hand figures is divisible by 4.

Thus, 4 is an exact divisor of 264, 1284, 1368, and 7932.

4. Five is an exact divisor of any number whose right hand figure is 0 or 5.

Thus, 5 is an exact divisor of 360, 1795, 3810, and 7895.

5. Six is an exact divisor of any even number, the sum of whose digits is divisible by 3.

Thus, 6 is an exact divisor of 732, 534, 798, and 8226.

6. Eight is an exact divisor of a number, if the number expressed by its three right hand figures is divisible by 8.

Thus, 8 is an exact divisor of 4328, 3856, 61360, and 5920:

7. Nine is an exact divisor of any number, the sum of whose digits is divisible by 9.

Thus, 9 is an exact divisor of 513, 1314, 252, 1341, and 312462.

8. 10, 100, 1000, etc., are exact divisors of any numbers that end respectively with one, two, three, etc., ciphers.

Thus, 10, 100, 1000, etc., are exact divisors respectively of 80, 800, 8000, etc.

9. If an even number is divisible by an odd number it is divisible by twice that number.

Thus, 72 is divisible by 9 and by twice 9 or 18. 312 by 3 and 6.

10. An exact divisor of a number is an exact divisor of any number of times that number.

Thus, 3 is an exact divisor of 12, and of any number of times 12, as 36.

11. An exact divisor of each of two numbers is an exact divisor of their sum and of their difference.

Thus, 3 is an exact divisor of 9 and 12 respectively, and therefore of 9+12, or 21; of 12-9, or 3.

117. Find by inspection some of the exact divisors of the following numbers:

FACTORING.

118. 1. What are the factors of 6? 8? 12? 16?

2. What factors of 18 are prime numbers or prime factors? 3. What are the prime factors of 30?

4. What are all the exact divisors of 30?

5. What numbers besides the prime factors of 30 are its exact divisors? How are they obtained from the prime

factors?

6. Of what number are 2, 3, and 5, the prime factors? 7. How can a number be obtained from its prime factors? 8. The prime factors of a number are 2, 2, and 5. What is the number? Give all the exact divisors of this number. 9. What are the exact divisors of 60? 72? 96? 144?

DEFINITIONS.

119. Factoring is the process of separating a number into its factors.

120. Prime Factors are factors that are prime numbers.

121. The number of times a number is used as a factor is indicated by a small figure called an exponent. It is written above and at the right of the number.

Thus, 4×4×4=43, and the 3 indicates that 4 is used as a factor three times.

122. PRINCIPLES.-1. Every prime factor of a number is an exact divisor of that number.

2. The only exact divisors of a number are its prime factors or the product of two or more of them.

3. Every number is equal to the product of its prime factors.

1. What are the prime factors of 756?

PROCESS.

2)756

2)378 3)189

ANALYSIS. Since every prime factor of a number is an exact divisor of the number, we may find the prime factors of 756 by finding all the prime numbers that are exact divisors of 756. Since the number is even, we divide by 2. Since the quotient obtained is 3)63 an even number, we divide again by 2. Then we divide by the prime numbers 3, 3, 3, successively, and the last quotient is 7, which is a prime number. 7 Hence the prime factors are 2, 2, 3, 3, 3, 7, or 22, 33, 7.

3)21

RULE.-Divide the given number by any prime number that will exactly divide it. Divide this quotient by another prime number, and so continue until the quotient is a prime number.

The several divisors and last quotient will be the prime factors.

48? 56? 63? 72?

4. When a number is multiplied by 4 and the product by 6, by what is the number multiplied?