The 1. c. m. must contain all of the prime factors of 414, 408, 3330, and each factor must occur as often in the 1. c. m. as in any one of the numbers. Thus, 3 must occur twice in the 1. c. m., 2 must occur three times, and 23, 17, 5, 37 must each occur once.

Therefore, the 1. c. m. = 2 × 2 × 2 × 3 × 3 × 5 × 23 × 17 × 37 = 5208120.

59. When the numbers cannot readily be factored, the g. c. d. may be used in finding the 1. c. m.

Since the g. c. d. contains all of the factors that are common to the numbers, if the numbers are divided by the g. c. d., the quotients will contain all the factors that are not common. The 1. c. m. is therefore the product of the quotients and the g. c. d. of the numbers.

Ex. Find the 1. c. m. of 14482 and 32721.

The g. c. d. of 14482 and 32721 is 13.

14482 ÷ 13 = 1114.

.. the 1. c. m. of the two numbers is

1114 x 32721 =36451194.

EXERCISE 13

1. Find the 1. c. m. and g. c. d. of 384, 2112, 2496. 2. Find the 1. c. m. of 3, 5, 9, 12, 14, 16, 96, 128. 3. Find the g. c. d. and 1. c. m. of 1836, 1482, 1938, 8398, 11704, 101080, 138945.

L

4. Prove that the product of the g. c. d. and 1. c. m. of two numbers is equal to the product of the numbers.

5. What is the length of the longest tape measure that can be used to measure exactly two distances of 2916 ft. and 3582 ft. respectively?

6. Find the number of miles in the radius of the earth, having given that it is the least number that is divisible by 2, 3, 4, 5, 6, 9, 10, 11, 12.

CASTING OUT NINES

60. The check on arithmetical operations by casting out the nines was used by the Arabs. It is a very useful check, but fails to detect such errors as the addition of 9, the interchange of digits, and all errors not affecting the sum of the digits.

(Why?)

The remainder arising from dividing any number by 9 is the same as that arising from dividing the sum of its digits by 9.

Thus, the remainder arising by dividing 75234 by 9 is 3, the same as arises by dividing 7 + 5 + 2 + 3 + 4 by 9.

The student should adapt the proof of Principle 5, p. 34, to this

statement.

61. The most convenient method is to add the digits, dropping or "casting out" the 9 as often as the sum amounts to that number.

Thus, to determine the remainder arising from dividing 645738 by 9, say 10 (reject 9), 1, 6, 13 (reject 9), 4, 7, 15 (reject 9), 6. Therefore, 6 is the remainder. After a little practice the student will easily group the 9's. In the above, 6 and 3, 4 and 5, could be dropped, and the excess in 7 and 8 is seen to be 6 at once.

62. Check on Addition by casting out the 9's.

Ex. Add 56342, 64723, 57849, 23454 and check the work by casting out the 9's.

Since each number is a multiple of 9 plus some remainder, the numbers can be written as indicated in the annexed solution.

563429 × 6260+ 2 rem. 647239 x 7191 + 4 rem. 57849 = 9 x 6427+ 6 rem. 234549 x 2606+ 0 rem.

2023689 × 22484 + 12 rem.

Thus, the excess of 9's is 3 and the excess in the sum of the excesses, 2, 4, 6, and 0, is 3, therefore the work is probably correct.

9x + r

9x' +' 9 x' +'

63. The proof may be made general by writing the numbers in the form 9x+r. This can be done since all numbers are multiples of 9 plus a remainder. Hence, by expressing the numbers in this form and adding we have for the sum a multiple of 9 plus the sum of the remainders. Therefore, the excess of the 9's in the sum is equal to the excess in the sum of the excesses.

9(x + x' + x' +•••) +r+r+r+ ...)

64. Check on Multiplication by casting out the 9's.

Since any two numbers may be written in the form 9x+r and

9 x + r

9 x' + r'

9 xr' + rr' 81xx' + 9 x'r

9x'r', multiplying 9x+r by 9 x'+r', we have 81 xx' + 9(x'r + xr')+rr'. From this it is evident that the excess of 9's in the product arises from the excess in rr'. Therefore, the excess of 9's in any product is

81 xx' + 9(x'r + xr')+r equal to the excess in the product found by multiplying the excesses of the factors together.

Ex. Multiply 3764 by 456 and check by casting out

The excess of 9's in 3764 is 2; the excess in 456 is 6; the excess in the product of the excesses is 3 (2 × 6 = 12; 12 – 9 = 3); the excess in 1716384, the product of the numbers, is 3. Therefore, the work is probably correct.

65. Check on Division by casting out the 9's.

Division being the inverse of multiplication, the dividend is equal to the product of the divisor and quotient plus the remainder. Therefore, the excess of 9's in the dividend is equal to the excess of 9's in the remainder plus the excess in the product found by multiplying the excess of 9's in the divisor by the excess of 9's in the quotient.

Ex. Divide 74563 by 428 and check by casting out the 9's. 74563 ÷ 428 = 174 + √28,

or

91

74563 = 174 × 428 + 91

The excess of 9's in 74563 is 7; in 174, 3; in 428, 5; in 91, 1. Since 7, the excess of 9's in 74563 = the excess in 3 x 5 + 1, or 16, which is the product of the excesses in 174 and 428 plus the excess in 91, the work is probably correct.

EXERCISE 14

1. State and prove the check on subtraction by casting out the 9's.

2. Determine without adding whether 89770 is the sum of 37634 and 52146.

3. Add 74632, 41236, 897321 and 124762, and check by casting out the 9's.

4. Multiply 76428 by 5937, and check by casting out the 9's.

5. Determine without multiplying whether 2718895 is the product of 3785 and 721.

6. Show by casting out 9's that 18149 divided by 56= 324.

7. Show that results may also be checked by casting out 3's; by casting out 11's.

8. Is 734657 divisible by 9? by 3? by 11?

9. Perform the following operations and check: 91728 x 762; 8496312463; 17 x 3.1416; 78.54 3.1416.

10. Does the proof for casting out the 9's hold as well for 4, 6, 8, etc.? May we check by casting out the 8's? Explain.

MISCELLANEOUS EXERCISE 15

1. What is the principle by which the ten symbols, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, are used to represent any number?

2. Why is the value of a number unaltered by annexing zeros to the right of a decimal?

3. How is the value of each of the digits in the number 326 affected by annexing a number, as 4, to the right of it? to the left of it?

4. How is the value of each of the digits of 7642 affected if 5 is inserted between 6 and 4?

5. Write 4 numbers of 4 places each that are divisible by (a) 4, (b) 2 and 5, (c) 6, (d) 8, (e) 9, (ƒ) 11, (g) 16, (h) 12, (i) 15, (j) 18, (k) 3, (1) 50, (m) 125, (n), both 6 and 9, (0) both 8 and 3, (p) both 30 and 20.

6. Determine the prime factors of the following numbers: (a) 3426, (b) 8912, (c) 6600, (d) 6534, (e) 136125, (f) 330330, (g) 570240.

7. Mr. Long's cash balance in the bank on Feb. 20 is $765.75. He deposits, Feb. 21, $150; Feb. 25, $350.25; Feb. 26, $97.50; and withdraws, Feb. 23, $200; Feb. 24, $123.40 and $112.50; Feb. 28, $321.75. What is his balance March 1?