FACTORING NUMBERS.

113. The FACTORS of a number are those numbers whose continued product is the number; thus, 3 and 7 are the factors of 21; 3 and 6, or 3, 3, and 2 are the factors of 18; etc.

NOTE 1. Every number is a factor of itself, the other factor being 1.

The prime factors of a number are those prime numbers whose continued product is the number; thus, the prime factors of 12 are 2, 2, and 3; the prime factors of 36 are 2, 2, 3, and 3; etc.

NOTE 2. Since 1, as a factor, is useless, it is not here enumerated.

114. To factor a number is to resolve or separate it into its factors. In resolving a number into its factors,

The following facts will be found convenient:

(a) Every number whose unit figure is 0, or an even number, is itself even, and .. divisible by 2.

(b) Any number is divisible by 3 when the sum of its digits (Art. 7) is divisible by 3; thus, 4257 is divisible by 3 because the sum of its digits, 4+2 +5 +7 = 18, is divisible by 3.

(c) Any number is divisible by 4 when 4 will divide the number expressed by the two right-hand figures; thus, 4 will divide 32,.. it will divide 7532.

(d) Any number whose unit figure is 0 or 5 is divisible by 5; as 90, 1740, 35, 34975, etc.

(e) Any even number which is divisible by 3 is also divisible by 6; thus, 3528 is divisible by 3 and.. by 6.

NOTE 1. For 7 no general rule is known.

(f) Any number is divisible by 8 when 8 will divide the num、 ber expressed by the three right-hand figures; thus, 8 will divide 816, .. it will divide 175816.

113. What are the Factors of a number? Is a number a factor of itself? What are the prime factors of a number? 114. What is it to factor a number? What number is divisible by 2? By 3? 4? 5? 6? What is said of 7? What number is divisible by 8?

(g) Any number is divisible by 9 when the sum of its digits is divisible by 9; thus, 7146 is divisible by 9 because the sum of its digits, 7+1+4+6=18, is divisible by 9.

(h) Any number ending with 0 is divisible by 10.

(i) Any number is divisible by 11 when the sum of the digits in the odd places is equal to the sum of the digits in the even places; also when the difference of these sums is divisible by 11; thus, 8129, in which 9+1=2+8, is divisible by 11; also 6280714, in which the sum of the digits in the odd places, 4+7 +8+ 6, differs from the sum of the digits in the even places, 1+0+2, by 22, a number divisible by 11.

(j) Any number divisible by 3 and also by 4, is divisible by 12; and, generally, any number that is divisible by each of several numbers that are mutually prime, is divisible by the product of those numbers; thus, 84 is divisible by 2, 3, and 7, separately, and .. 84 is divisible by 2 × 3 × 7 = 42; so also 108 is divisible by 4 and 9, and .. by 4 × 9 = 36.

NOTE 2. Every prime number, but 2 and 5, has 1, 3, 7, or 9 for its unit figure.

For further aid in determining the factors of numbers, we have the following

TABLE OF PRIME NUMBERS FROM 1 TO 997.

1 41 101 167 239 313 397 467 569 643 733 823 911 2 43 103 173 241 317 401 479 571 647 739 827 919 3 47 107 179 251 331 409 487 577 653 743 829 929 5 53 109 181 257 337 419 491 587 659 751 839 937 7 59 113 191 263 347 421 499 593 661 757 853 941 11 61 127 193 269 349 431 503 599 673 761 857 947 13 67 131 197 271 353 433 509 601 677 769 859 953 17 71 137 199 277 359 439 521 607 683 773 863 967 19 73 139 211 281 367 443 523 613 691 787 877 971 23 79 149 223 283 373 449 541 617 701 797 881 977 29 83 151 227 293 379 457 547 619 709 809 883 983 31 89 157 229 307 383 461 557 631 719 811 887 991 37 97 163 233 311 389 463 563 641 727 821 907 997

114. What number is divisible by 9? By 10? 11? 12?

General principle?

115. A PROBLEM is something to be done; or, it is a question which requires a solution. The solution of a problem consists of the operations necessary for finding the answer to the question. To solve a problem is to perform the operations for finding the answer.

116. PROBLEM 1. To resolve or separate a number into its prime factors:

RULE. Divide the given number by any prime number greater than one, that will divide it; divide the QUOTIENT by any prime number greater than one that will divide IT, and so on till the quotient is prime. The several divisors and last quotient will be the prime factors sought.

Ex. 1. What are the prime factors of 30? Ans. 2, 3, and 5.

OPERATION.

2)30 3)15

5

It is immaterial in what order the prime factors are taken, though it will usually be most convenient to take the smaller factors first.

Ans. 2, 2, 2, and 3.
Ans. 2, 2, 3, and 7.
Ans. 3, 5, 5, and 5.

2. What are the prime factors of 24? 3. Resolve 84 into its prime factors. 4. Resolve 375 into its prime factors. 5. What are the prime factors of 3465? 6. What are the prime factors of 19800? 7. What are the prime factors of 1440 ? 8. What are the prime factors of 3150? 9. What are the prime factors of 2310? 10. What are the prime factors of 1728? 11. What are the prime factors of 1800? 12. What are the prime factors of 2448? 13. What are the prime factors of 4824? 14. What are the prime factors of 3648? 15. What are the prime factors of 8696? 16. What are the prime factors of 7264?

17. What are the prime factors of 5075?

115. What is a Problem? The solution of a problem? What is it to solve a

problem? 116. Rule for finding the prime factors of a number?

117. If a number has composite factors, they may be found by multiplying together two or more of its prime factors; thus, the prime factors of 12 are 2, 2, and 3, and the composite factors of 12 are 2 X 2, 2 X 3, and 2 X 2 X 3, i. e. the composite factors of 12 are 4, 6, and 12.

GREATEST COMMON DIVISOR.

118. A COMMON DIVISOR of two or more numbers is any number that will divide each of them without remainder; thus 3 is a common divisor of 12, 18, and 30.

119. The GREATEST COMMON DIVISOR of two or more numbers is the greatest number that will divide each of them without remainder; thus, 6 is the greatest common divisor of 12, 18, and 30.

NOTE. A divisor of a number is often called a measure of the number, also an aliquot part of the number.

120. PROBLEM 2. To find the greatest common divisor of two or more numbers.

Ex. 1. What is the greatest common divisor of 18, 30, and 48?

OPERATION.

18 =2X 3X 3

30

2 X 3 X 5

48

2 × 3 × 2 × 2 × 2

uct, 2 X 3 numbers.

Ans. 2 X 3 6.

We see that 2 and 3 are factors common to all the numbers, and, furthermore, they are the only common factors; hence their prod

6, is the greatest common divisor of the given

2. What is the greatest common divisor of 60, 72, 48, and 84?

117. Composite factors, how formed?

Ans. 2 X 2 X 3=12.

Although 2 is a factor more than twice in some of the given numbers, yet, as it is a factor only twice in others, we are not at liberty to take 2 more than twice

118. What is a Common Divisor?

119. Greatest Common Divisor? Other names for divisor?

in finding the greatest common divisor. The same remark applies to other factors.

Hence,

RULE 1. Resolve each number into its prime factors, and the continued product of all the prime factors that are common to all the given numbers will be the common divisor sought.

3. What is the greatest common divisor of 24, 40, 64, 80, 96, 120, and 192? Ans. 2 X 2 X 2 8. 4. Find the greatest common divisor of 15, 45, 75, 105, 135, 150, and 300.

Ans. 15.

5. Find the greatest common divisor of 25, 45, and 70.

Ans. 5.

6. Find the greatest common divisor of 24, 36, and 64.

Ans. 4.

7. Find the greatest common divisor of 24, 48, 72, and 88. 8. Find the greatest common divisor of 45, 75, 90, 135, 150, and 180.

9. I have three rooms, the first 11ft. 3in. wide, the second 15ft. 9in. wide, and the third 18ft. wide; how wide is the widest carpeting which will exactly fit each room? How many breadths will be required to cover each room?

1st Ans. 27 inches.

121. When the given numbers are not readily resolved into their prime factors, their greatest common divisor may be more easily found by

RULE 2. Divide the greater of two numbers by the less, and, if there be a remainder, divide the divisor by the remainder, and continue dividing the last divisor by the last remainder until nothing remains; the last divisor is the greatest common divisor of the two numbers.

If more than two numbers are given, find the greatest divisor of two of them, then of this divisor and a third number, and so on until all the numbers have been taken; the last divisor will be the divisor sought.

120. Rule for finding the greatest common divisor of two or more numbers? 121. Second rule for finding greatest common divisor?