thus when N and D are divided by r+1 the sum of the remainders must be r+ 1, unless either remainder is zero, and then the other remainder also is zero.

For example, suppose r = 10 and N=263419. Here

9-1+4 −3+ 6 − 2 = 13 =D;

and N and D when divided by 11 each leave the remainder 2.

Again, suppose r = 10 and N=615372. Here

2-7+3-5+1-6=-12=-D;

and N and D when divided by 11 leave the remainders 10 and 1 respectively.

445. To find what numbers are divisible by 3 without remainder.

72

Let N denote any number; let Po, PP be the digits of it beginning with that in the unit's place; then

N=p2+p ̧10+ p ̧102 + +p,10";

......

This is a whole number when Po+P1+P2

3

number. Thus any number is divisible by 3 when the sum of its digits is divisible by 3. For example, 111, 252, and 7851 are divisible by 3.

446. It appears from Art. 443 that a number is divisible by 9 when the sum of its digits is divisible by 9; and that when any number is divided by 9, the remainder is the same as if the sum of the digits of that number were divided by 9.

It appears from Art. 444 that a number is divisible by 11 when the difference between the sum of the digits in the odd

places and the sum of the digits in the even places is divisible by 11.

447. From the property of the number 9, mentioned in the preceding article, a rule may be deduced which will sometimes detect an error in the multiplication of two numbers.

Let 9a+x denote the multiplicand, and 96 + y the multiplier; then the product is 81ab + 9bx +9ay + xy. If then the sum of the digits in the multiplicand be divided by 9, the remainder is x; if the sum of the digits in the multiplier be divided by 9, the remainder is y; and if the sum of the digits in the product be divided by 9, the remainder ought to be the same as when xy is divided by 9, and will be if there be no mistake in the operation.

EXAMPLES ON SCALES OF NOTATION.

1. Express in the scale of seven the numbers which are expressed in the scale of ten by 231 and 452; multiply the numbers together in the scale of seven, and reduce to the scale of ten.

2. Transform 1357531 from the denary scale to the quinary. 3. Transform 40234 from the quinary to the duodenary scale.

4. Transform 545 from the senary scale to the denary.

5. Transform 64520, which is in the septenary scale, to the undenary scale.

6. Transform 4444 from the scale with radix five to the common scale.

7. Transform 3413 from a scale whose radix is six to that whose radix is seven.

8. Transform 123456 from the denary scale to the septenary. 9. Transform 15.75 from scale ten to scale eight.

10. Transform 221.248 from scale ten to scale five.

11. Express 357234 in the scale whose radix is seven.

12. Transform 1845.3125 from the common scale to one whose radix is twelve.

13. Transform 444.44 from the scale with radix five to the common scale.

14. Express 31462-125 in the scale whose radix is eight.

15. Transform 3065.263 from scale eight to scale ten.

16. Express in the common scale and in the scale of eight the number denoted in the scale of nine by 723.

17. Transform 15951 from scale eleven to scale ten, and 333310 from scale ten to scale eleven.

18. Extract the square root of 33224 in the scale of six.

19. The number 123454321 is referred to the radix six; extract its square root in that scale.

20. Extract the square root of 3445 44 in the scale six, and reduce the result to scale three.

21. Subtract 20404020 from 103050301 in the scale eight, and extract the square root of the result.

22. Extract the square root of 11000000100001 in the binary scale of notation.

23. Find a fraction in the ternary scale equivalent to 120120......, which is in the same scale.

24. Find the simplest fraction which is represented by 1515...... in the scale whose radix is seven.

26.

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In what scale will the number 95 be denoted by 137?

27.

In what scale is 2704 written 20304?

28. In what scale is 1331 written 1000?

29. In what scale does 16000 of the denary become 1003000 ?

30. A number is represented in the denary scale by 35.8333..., and in another scale by 55.5, find the radix of the latter scale.

31. In what scale of notation is sixteen hundred and sixtyfour ten-thousandths of unity represented by '0404?

32. Shew that 12345654321 is divisible by 12321 in any scale greater than six.

33. Shew that 144 is a square number whatever be the radix of the scale; the radix being supposed greater than four.

34. Shew that 1331 is a perfect cube in any scale of notation; the radix being supposed greater than three.

35. Of the weights 1, 2, 4, 8,......2" pounds, find which must be selected to weigh 1719 pounds.

36. Which of the weights 1 lb., 3 lbs., 3 lbs.,...... must be selected to weigh 1027 lbs., not more than one of each kind being used, but in either scale that is necessary?

37. Which of the same weights must be used to weigh 716 lbs.?

38. Which of the same weights must be used to weigh 475 lbs.?

39. Find by operation in the scale with radix twelve what is the height of a parallelepiped which contains 94 cubic feet 235 cubic inches, and whose base is 24 square feet 5 square inches.

40. Express 2 feet 10 inches linear measure, and 5 feet 791 inches square measure, in the duodenary scale as feet and duodecimals of a foot; and the latter quantity being the area of a rectangle, one of whose sides is the former, find its other side by dividing in the duodenary scale.

41. If Po, P1, P... be the digits of a number beginning with the units, prove that the number itself is divisible by eight if p ̧+2p, + 4p, is divisible by eight.

1

2

42. Prove that the difference of two numbers consisting of the same figures is divisible by nine.

43. Find the greatest and least numbers with a given number of digits in any proposed scale.

44. Prove that if in any scale of notation the sum of two numbers is a multiple of the radix, then (1) the digits in which the squares of the numbers terminate are the same, and (2) the sum of this digit and of the digit in which the product of the numbers terminates is equal to the radix.

45. A certain number when represented in the scale two has each of its last three digits (counting from left to right) zero, and the next digit different from zero; when represented in either of the scales three, five, the last digit is zero, and the last but one different from zero; and in every other scale (twelve scales excepted) the last digit is different from zero. What are these twelve scales, and what is the number?

XXX. ARITHMETICAL PROGRESSION.

448. Quantities are said to be in Arithmetical Progression when they increase or decrease by a common difference.

Thus the following series are in Arithmetical Progression:

In the first example the common difference is 2, in the

second -4, in the third b, in the fourth-b.