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(h.) Two numbers are PRIME TO EACH OTHER when they have no common factors.

ILLUSTRATIONS. — 4 and 9 are prime to each other; but 6 and 9 are not, because they have the common factor 3.

(i.) A number is divided into factors when any factors which will produce it are found.

ILLUSTRATIONS. — 12 = 2 X 2 X 3; 30 = 2 X 3 X 5.

(j.) The product of a number taken any number of times as a factor is called a POWER of that number.

ILLUSTRATION. —8, which is the product of 2 X 2 X 2, i. e. of 2 taken 3 times as a factor, the third power of 2.

(k.) We may indicate the power of a number by writing a little figure called an EXPONENT above it and a little to the rigbt.

ILLUSTRATIONS. —"33" is read 3 to the third power, and equals 3 X 3 X 3, or 27; “25” is read 2 to the fifth power, and equals 2 X 2 X 2 X 2 X 2.

(1.) The second power of a number is sometimes called its SQUARE, and the third power its CUBE.

ILLUSTRATIONS. —8, or 2, is the cube of 2; 25, or 5”, is the square of 5.

69. Properties of Numbers.

(a.) A number which is a factor of another number must also be a factor of any multiple of that other number.

ILLUSTRATION. — 4 is a factor of 12, and hence of 24, 36, 48, or any other multiple of 12.

(b.) If each of two numbers is divisible by a third number, both their sum and their difference must be divisible by that third number.

ILLUSTRATION. – Both 12 and 30 are divisible by 6. Hence, their sum, 42, and their difference, 18, are also divisible by 6.

(c.) If one of two numbers is divisible by a third number, and the other is not, neither their sum nor their difference will be divisible by that third number.

ILLUSTRATION.-12 is divisible by 4 and 21 is not; hence, neither their sum, 33, nor their difference, 9, is divisible by 4.

(d.) If neither of two numbers is divisible by a third, their sum and their difference may or may not be divisible by that third number.

ILLUSTRATIONS.— Neither 10 nor 18 are divisible by 4; yet both their sum, 28, and their difference, 8, are divisible by 4.

Again: neither 11 nor 25 are divisible by 4; yet their sum, 36, is, and their difference, 14, is not divisible by 4.

(e.) If a number divisible by each of two numbers which are prime to each other, it must also be divisible by their product.

ILLUSTRATION. A number which is divisible by 4 and 9, must be divisible by 36.

(f.) If a number is divisible by each of two numbers which have a common factor, it may or may not be divisible by their product.

ILLUSTRATIONS. -80 is divisible by 4 and 10, and by their product, 40; 140 is divisible by 4 and 10, but not by their product, 40.

(g.) A number which is not divisible by another will not be divisible by any multiple of that other number.

ILLUSTRATION. - A number which is not divisible by 2, is not divisible by 4, 6, 8, 10, or any other multiple of 2.

(h.) A number which is divisible by any composite number, is also divisible by all the factors of that composite number.

ILLUSTRATION.—A number which is divisible by 36, is divisible by 2, 3, 4, 6, 9, 12, and 18, the factors of 36.

(i.) A number is divisible by 2 or 5, when its right-hand figure is thus divisible.

ILLUSTRATIONS. — 470 is divisible by 2 and by 5, because the right-hand figure is zero; 538 is divisible by 2 because 8 is.

(j.) A number is divisible by 4, 20, 25, or 50, when the number expressed by its two right-hand figures is thus divisible.

ILLUSTRATION. — 1724 is divisible by 4 because 24 is; but is not divisible by 20, 25, or 50, because 24 is not.

(k.) A number is divisible by 8, 40, 125, 250, or 500, when the number expressed by its three right-hand figures is thus divisible.

ILLUSTRATION. - 54750 is divisible by 125 and 250 because 750 is; but is not divisible by 8, 40, or 500, because 750 is not.

(1.) A number is divisible by 3 or by 9, when the sum of its digit figures is thus divisible.

ILLUSTRATIONS.—The sum of the digit figures of 2574 is 2 + 5 + 7 + 4 18, which is divisible by 9; hence, 2574 is divisible by 9. The sum of the digits of 8571 is 8 + 5 + 7+1 = 21, which is divisible by 3, but not by 9; hence, 8571 is divisible by 3, but not by 9.

(m.) A number is divisible by 11 when the sums of its alternate digits are alike, or their difference is a multiple of 11.

ILLUSTRATIONS. — In 857934, the sum of the first set of alternate digits is 4 + 9 + 5 = 18; the sum of the second set is 3 + 7 + 8 = 18. These sums being alike, the number is divisible by 11. In 8295716, the sum of the first set of alternate digits is 6 + 7 + 9 + 8 = 30; the sum of the second set is 1 + 5 + 2 = 8. Their difference, 30 8 = 22, being a multiple of 11, the number is divisible by 11. In 716896, the sum of the first set of alternate digits is 6 + 8 + 1 = 15; the sum of the second set is 9 + 6 +7 – 22. Their difference, 22 — 15 = 7, not being a multiple of 11, the number is not divisible by 11.

70. Exercises in Factoring Numbers.

1. What are the prime factors of 72?

Solution. — 72 = 8 X 9, which, since 8 = 28 and 9 = 3?, gives 72 - 23 X 3.

Find the prime factors of the following numbers :
2. 36.
5. 63.

8. 84.
3. 24.
6. 27.

9. 96.
4. 81.
7. 48.

10. 35.

11. 99.
12. 54.
13. 30.

14. What are the prime factors of 3564?

Solution. - It is readily seen that this number is divisible by 4, 9, and 11, and hence (see 69, e) by their product, 396. Dividing by 396 gives 9 for a quotient. Hence 3564 = 4 X 9 X 11 X 9, or since 4 = 22, and 9

3', 3564 = 22 X 3' X 11 X 32 = 22 X 3 X 11.

15. What are the prime factors of 381191 ?

SOLUTION.—We first observe that 381191 is not divisible by 2, 3, 5, or 11, and hence that it is not divisible by any multiple of these numbers. We * This conclusion depends on the fact that the larger the divisor is, the smaller will be the quotient. Hence, any number larger than 37 must give a quotient less than that obtained by 37, 1. e. less than 35, and as we have found that no number less than 37 is a factor of 1319, it follows that none greater than 37 can be.

have, then, to try other prime numbers, beginning, for convenience, with the smallest. By actual trial, we find that 7 and 13 each leave a remainder after division; but that 17 is contained in it 22423 times. We have now to find the factors of 22423. It can contain no factor less than 17, for the original number contained none. We therefore try 17, and find that it is contained 1319 times. We have, therefore, to find the factors of 1319. Beginning with 17, we try in succession the prime numbers 17, 19, 23, 29, 31, and 37, each of which leaves a remainder. But dividing by 37 gives 35 for the entire part of the quotient, which, being less than the divisor, 37, shows that 1319 is a prime number.* Hence, the prime factors required are 17 X 17 X 1319 17" X 1319.

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(a.) A Divisor of a number is any number which will exactly divide it.

(b.) A Common Divisor of two or more numbers is any number which is a divisor of each of them.

(c.) The GREATEST COMMON Divisor of two or more numbers is the largest number which is a divisor of each of them.

(d.) From these definitions, and the principles previously illustrated, it is obvious that

1st. A Divisor of a number is the product of prime factors found in that number.

2d. A Common Divisor of two or more numbers is the product of prime factors which are common to all those numbers.

3d. The Greatest Common Divisor of two or more numbers is the product of all the prime factors common to those numbers.

(e.) To find the Greatest Common Divisor of two or more numbers, we factor one of the numbers, and then see which of its prime factors, if any, are factors of all the other given numbers. The product of the prime factors thus found is the Common Divisor sought.

1. What is the greatest common divisor of 36, 54, and 60 ?

SOLUTION. — 36 = 22 X 32. Of these factors, 54 contains only 2 and 3%; and of these last, 60 contains only 2 and 3. Hence, the only common factors are 2 and 3, and the greatest common divisor must be 2 X 3, or 6.

2. What is the greatest common divisor of 46, 69, and 72?

SOLUTION. — 46 = 2 X 23. But 2 cannot be a factor of the common divi. sor, because it is not a factor of 69, and 23 cannot be, because it is not a factor of 72. Hence, the only factor common to all the numbers is 1, and 1 is the greatest common divisor.

3. What is the greatest common divisor of 378, 504, 567, and 756?

SOLUTION. — 504 = 28 X 39 X 7. But 2 cannot be a factor of the greatest common divisor, because it is not a factor of 567. Next considering 38 and 7, we find that they are factors of all the numbers. Hence, the greatest common divisor must be 32 X 7, or 63.

What is the greatest common divisor of

4. 12 and 18? 5. 15 and 25? 6. 24 and 36 ?

7. 30 and 42? 8. 51 and 102 ? 9. 28 and 42?

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