If in the successive multiplications by r any one of the products is an integer the process terminates at this stage, and the given fraction can be expressed by a finite number of digits. But if none of the products is an integer the process will never terminate, and in this case the digits recur, forming a radixfraction analogous to a recurring decimal. Example 2. Transform 16064.24 from scale eight to scale five. After this the digits in the fractional part recur; hence the required number is 212340-1240. 82. In any scale of notation of which the radix is r, the sum of the digits of any whole number divided by r-1 will leave the same remainder as the whole number divided by r - 1. Let N denote the number, a, a, a,,......a, the digits beginning with that in the units' place, and S the sum of the digits; then n-1 S= a。 + a2+ a2 + ...... ...N-S-a, (r-1) + a ̧ (x2 - 1) + ......+a, (?" — 1) + α (2′′ − 1). = 2 Now every term on the right hand side is divisible by r where I is some integer; which proves the proposition. Hence a number in scaler will be divisible by r 1 when the sum of its digits is divisible by r – 1. 83. By taking_ r = 10 we learn from the above proposition that a number divided by 9 will leave the same remainder as the sum of its digits divided by 9. The rule known as 66 casting out the nines" for testing the accuracy of multiplication is founded on this property. The rule may be thus explained: Let two numbers be represented by 9a+b and 9c+d, and their product by P; then P=81ac + 9bc + 9ad + bd. P bd ; and therefore the sum of the digits of P, when divided by 9, gives the same remainder as the sum of the digits of bd, when divided by 9. If on trial this should not be the case, the multiplication must have been incorrectly performed. In practice b and d are readily found from the sums of the digits of the two numbers to be multiplied together. Example. Can the product of 31256 and 8427 be 263395312 ? The sums of the digits of the multiplicand, multiplier, and product are 17, 21, and 34 respectively; again, the sums of the digits of these three numbers are 8, 3, and 7, whence bd=8x3=24, which has 6 for the sum of the digits; thus we have two different remainders, 6 and 7, and the multiplication is incorrect. 84. If N denote any number in the scale of r, and D denote the difference, supposed positive, between the sums of the digits in the odd and the even places; then N-D or N+D is a multiple of r + 1. n Let a, a, a,,... a denote the digits beginning with that in the units' place; then N = a ̧ + a ̧r+a ̧12 + α ̧μ3 + ...... +α-¿?"~' + a ̧?”. 2 2 n a_r". 3 = a, (r + 1) + a ̧ (22 − 1) + α ̧ (2o3 + 1) + and the last term on the right will be a (r” + 1) or a„ (?” − 1) according as n is odd or even. Thus every term on the right is divisible by r+ 1; hence COR. If the sum of the digits in the even places is equal to the sum of the digits in the odd places, D = 0, and N is divisible by r + 1. Example 1. Prove that 4:41 is a square number in any scale of notation whose radix is greater than 4. Example 2. In what scale is the denary number 2.4375 represented by Sometimes it is best to use the following method. Example 3. In what scale will the nonary number 25607 be expressed by 101215? The required scale must be less than 9, since the new number appears the greater; also it must be greater than 5; therefore the required scale must be 6, 7, or 8; and by trial we find that it is 7. Example 4. By working in the duodenary scale, find the height of a rectangular solid whose volume is 364 cub. ft. 1048 cub. in., and the area of whose base is 46 sq. ft. 8 sq. in. The volume is 3641948 cub. ft., which expressed in the scale of twelve is 264.734 cub. ft. The area is 46 sq. ft., which expressed in the scale of twelve is 3t.08. We have therefore to divide 264.734 by 3t.08 in the scale of twelve. 6. 7. 8. Transform 212231 from scale four to scale five. Express the duodenary number 398e in powers of 10. 9. Transform 213014 from the senary to the nonary scale. 11. Transform 400803 from the nonary to the quinary scale. Express the septenary number 20665152 in powers of 12. Transform ttteee from scale twelve to the common scale. 12. 13. 3 10 14. Express as a radix fraction in the septenary scale. 15. Transform 17.15625 from scale ten to scale twelve. 16. Transform 200-211 from the ternary to the nonary scale. 17. Transform 71·03 from the duodenary to the octenary scale. 18. 1552 2626 Express the septenary fraction as a denary vulgar fraction in its lowest terms. 19. Find the value of 4 and of 42 in the scale of seven. 20. In what scale is the denary number 182 denoted by 222? 25 21. In what scale is the denary fraction denoted by 0302? 128 Н. Н. А. 5 22. Find the radix of the scale in which 554 represents the square of 24. 23. In what scale is 511197 denoted by 1746335? 24. Find the radix of the scale in which the numbers denoted by 479, 698, 907 are in arithmetical progression. 25. In what scale are the radix-fractions 16, 20, 28 in geometric progression? 26. The number 212542 is in the scale of six; in what scale will it be denoted by 17486? 27. Shew that 148.84 is a perfect square in every scale in which the radix is greater than eight. 28. Shew that 1234321 is a perfect square in any scale whose radix is greater than 4; and that the square root is always expressed by the same four digits. 29. Prove that 1.331 is a perfect cube in any scale whose radix is greater than three. 30. Find which of the weights 1, 2, 4, 8, 16,... lbs. must be used to weigh one ton. 31. Find which of the weights 1, 3, 9, 27, 81,... lbs. must be used to weigh ten thousand lbs., not more than one of each kind being used but in either scale that is necessary. 32. Shew that 1367631 is a perfect cube in every scale in which the radix is greater than seven. 33. Prove that in the ordinary scale a number will be divisible by 8 if the number formed by its last three digits is divisible by eight. 34. Prove that the square of rrrr in the scale of s is rrrq0001, where q, r, s are any three consecutive integers. 35. If any number N be taken in the scale r, and a new number N' be formed by altering the order of its digits in any way, shew that the difference between N and N' is divisible by r−1. 36. If a number has an even number of digits, shew that it is divisible by r+1 if the digits equidistant from each end are the same. 37. If in the ordinary scale S be the sum of the digits of a number N, and 3S2 be the sum of the digits of the number 3Ñ, prove that the difference between S, and S2 is a multiple of 3. 38. Shew that in the ordinary scale any number formed by writing down three digits and then repeating them in the same order is a multiple of 7, 11, and 13. 39. In a scale whose radix is odd, shew that the sum of the digits of any number will be odd if the number be odd, and even if the number be even. 40. If n be odd, and a number in the denary scale be formed by writing down n digits and then repeating them in the same order, shew that it will be divisible by the number formed by the n digits, and also by 9090...9091 containing n - 1 digits. |