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A number is divisible by 8 if the number composed of the last three digits be so divisible.

A number is divisible by 9 if the sum of its digits, or the sum of the numbers formed by grouping the digits in pairs, be so divisible.

A number is divisible by 11 if the sum of the numbers formed by grouping the digits in pairs be so divisible. Thus 38764 is divisible by 11, for 64+87+3=154, and 154 is divisible by 11 because 54+1 or 55 is so divisible.

35. To resolve a number into its prime factors: Rule.-Divide the given number by any prime factor, divide the quotient in the same manner, and repeat the operation until the quotient is a prime number. The several divisors and the last quotient are the prime factors required.

Resolve 13860 into prime factors :

2)13860

2)6930

5)3465

11)693

7)63

3)9

3

The prime factors are 2, 2, 5, 11, 7, 3, 3, or arranging them in numerical order, 2, 2, 3, 3, 5, 7, 11.

EXERCISE 8.

Resolve into their prime factors

A. (1) 18; (2) 24; (3) 36; (4) 28; (5) 16; (6) 42; (7) 48; (8) 32; (9) 27; (10) 56.

B. (1) 54; (2) 63; (3) 80; (4) 96; (5) 81; (6) 99; (7) 105; (8) 75; (9) 125; (10) 132.

C. (1) 144; (2) 252; (3) 135; (4) 315; (5) 616; (6) 560; (7) 396; (8) 462; (9) 840; (10) 924. D. (1) 945; (2) 504; (3) 693; (4) 9240; (5) 3696; (6) 1925; (7) 4675; (8) 1035; (9) 17160; (10) 14280.

CHAPTER VIII.

GREATEST COMMON DIVISOR.

36. A common divisor of two or more numbers is a number that will exactly divide each of them.

37. The greatest common divisor (G.C.D.) of two or more numbers is the greatest number that will exactly divide each of them.

38. A common divisor is sometimes called a common measure, and the greatest common divisor the greatest common measure (G.C.M.).

39. If each of two numbers be separated into two factors one of which is common to both, and the other factors are prime to one another, the common factor is the G.C.D. of the given numbers.

Thus 15 and 35, or 3 times 5 and 7 times 5, have

a common factor 5, and the other factors 3 and 7 are prime to each other; therefore the common factor 5 is the G.C.D. of 15 and 35.

40. If the greater of two numbers be divided

;

by the less, the G.C.D. of the remainder (if any) and the divisor is also the G.C.D. of the divisor and dividend. If there be no remainder the divisor is the G.C.D. of the two numbers.

Thus, taking the numbers 18 and 48, and dividing, we get a remainder, 12. The division may be performed thus—

3 times 6)8 times 6(2

6 times 6

2 times 6

Now the remainder 12 and the divisor 18, or 2 times 6 and 3 times 6, have a common factor 6, and since the other factors are prime to one another, therefore 6 is the G.C.D. of the remainder and divisor. For a similar reason 6 is the G.C.D. of the divisor and dividend; therefore the G.C.D. of the remainder and divisor is also the G.C.D. of the divisor and dividend.

Find the G.C.D. of 27 and 72.

27)72(2

54

18

18)27(1

18

9)18(2

18

Explanation.-Dividing 72 by 27 we get the remainder, 18; ... the G.C.D. of 18 and 27 is the G.C.D. of 27 and 72. Again, taking 18 and 27, and dividing, we get a remainder 9. the G.C.D. of 9 and 18 is the G.C.D. of 18 and 27. Lastly, dividing 18 by 9, there is no remainder, and therefore 9 is the G.C.D. of 9 and 18; it is therefore the G.C.D. of 18 and 27, and therefore of 27 and 72.

The operation may be conveniently arranged as follows::

27)72(2

54

18)27(1
18

9)18(2
18

41. Rule.-Divide the greater number by the less, then the less number by the remainder, then the first remainder by the second remainder. Proceed in this way until there is no remainder, and the last divisor will be the G.C.D. required.

42. To find the G.C.D. of three or more numbers.

First find the G.C.D. of two of them, and then of this G.C.D. and one of the remaining numbers, and so on to the last number. The last divisor obtained is the G.C.D. of the given numbers.

EXERCISE 9.

Greatest Common Divisor.

A. Write down all the exact divisors of (1) 18; 28; (3) 36; (4) 42; (5) 54; (6) 60; (7) 63; (8) 68; (9) 72; (10) 84.

B. Write down all the common divisors of (1) 12 and 18; (2) 16 and 24; (3) 20 and 36; (4) 48 and 54; (5) 60 and 45; (6) 120 and 90; (7) 84 and 96; (8) 64 and 48; (9) 50 and 125; (10) 72

and 108.

C. Find by inspection the G.C.D. of (1) 45 and 54; (2) 60 and 90; (3) 42 and 56; (4) 64 and 24; (5) 48 and 120; (6) 28, 52, and 44; (7) 91,

63, and 49; (8) 54, 30, and 48; (9) 126, 162, and 144; (10) 180, 132, and 156.

Find by continual division or otherwise the G.C.D. of

D. (1) 799 and 663; (2) 1587 and 1334; (3) 1036 and 1288; (4) 1537 and 841; (5) 1855 and 4028; (6) 1890 and 2905; (7) 2077 and 2573; (8) 5133 and 2773; (9) 3151 and 2466; (10) 3346 and 5497.

E. (1) 2041 and 8476; (2) 3852 and 5265; (3) 4752 and 52866; (4) 18607 and 417979; (5) 49373 and 147731; (6) 428571 and 999999; (7) 7728, 8512, and 3456; (8) 4692, 7956, and 1044; (9) 7308, 18228, and 4998; (10) 7254, 20202, and 7917.

CHAPTER IX.

LEAST COMMON MULTIPLE.

43. When one number is exactly divisible by another, the first number is called a multiple of the other number. Thus 12 is a multiple of 3 because it is exactly divisible by 3.

44. A common multiple of two or more numbers is a number which is exactly divisible by each of them without a remainder. Thus 12 is a common multiple of 2, 3, 6.

45. The Least Common Multiple (L.C.M.) of two or more numbers is the least number which is exactly divisible by each of them without a remainder. Thus 6 is the L.C.M. of 2, 3, 6.

46. The L.C.M. of two or more numbers must contain all the prime factors of those numbers, and no other factors.

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