Numbers which have no common factor greater than unity, are said to be PRIME to one another. Thus the numbers 3, 5, 8, 11, are prime to each other. 45. When the divisor is a composite number, and made up of two factors, neither of which exceeds 12, the dividend may be divided by one of the factors in the way of Short Division, and then the result by the other factor: if there be a remainder after each of these divisions, the true remainder will be found by multiplying the second remainder by the first divisor, and adding to the product the first remainder. the total remainder is 9 × 3+5, or 27+5=32. Therefore the quotient arising from the division of 56732 by 45 is 1260, with a remainder 32 over. The reason for the above Rule is manifest from the following considerations. 6303 is 5 times 1260 together with 3, and 56732 is 9 times 6303 together with 5, or is 9 times (5 times 1260+3), together with 5, or is 45 times 1260+27 +5, or is 45 times 1260+32. 46. The accuracy of results in Multiplication is often tested by the following method, which is termed " CASTING OUT THE NINES": add together all the figures in the multiplicand, divide their sum by 9, and set down the remainder; then divide the sum of the figures in the multiplier by 9, and set down the remainder: multiply these remainders together, and divide their product by 9, and set down the remainder: if this remainder be the same as the remainder which results after dividing the product, or the sum of the digits in the product, of the multiplicand and multiplier by 9, the sum is very probably right; but if different, it is sure to be wrong. This test depends upon the fact that "if any number and the sum of its digits be each divided by 9, the remainders will be the same." The proof of which may be shewn thus: 100=99+1, where the remainder must be one, whether 100, or the sum of the digits in 100, viz. 1, be divided by 9, since 99 is divisible by 9 without a remainder. Hence it appears that if 100, 200, 300, 400, 500, &c. be each divided by 9, and the sum of the digits making up the respective numbers be also divided by 9, the two remainders in each case will be the same. Also the number 532=500+30+2 =5×100+3 × 10+2 =5×99+5+3×9+3+2; whence it appears that if the parts 5×100, 3×10, and 2, which make up the entire number, be each divided by 9, the remainders will be 5, 3, 2 respectively; and therefore the remainder, when 532 is divided by 9, will clearly be the same, as when 5+3+2 is divided by 9. To explain why the test holds, let us take as an example 533 multiplied by 57. It is clear, since 531 contains 9 without a remainder, that 531 × 57 contains 9 without a remainder; therefore the remainder which is left after dividing the product of 533 and 57 by 9, must be the same as the remainder which is left after dividing the product of 2 and 57 by 9. Again, since the product of 57 and 2=(54+3) × 2, and the product of 54 and 2 when divided by 9 leaves no remainder, therefore the remainder which is left after dividing the product of 533 and 57 by 9, must be the same as the remainder left after dividing the product of 3 and 2 by 9, i.e. after dividing the product of the remainders which are left after the division of the multiplicand and multiplier respectively by 9. Now on dividing either 30381, or the sum of its digits, which is 15, by 9, the remainder left is 6, and 3× 2 divided by 9 also leaves 6 as remainder. Therefore we conclude that 30381 is the correct product of 533 and 57. Note. If an error of 9, or any of its multiples, be committed, the results will nevertheless agree, and so the error in that case remains undetected. (1) 456÷2. (4) 6378÷3. (10) 9876540÷5. (13) 623399÷7. (15) 164864÷8. (17) 78692319. (19) 407792÷11. (21) 211632÷12. (23) 404586-13. (25) 1234560-20. (20) 91875342÷11. (22) 43600391÷12. (45) 10000000000000000÷1111, and also by 11111. (46) 634394567÷164600. (48) 1220225292 ÷ 200563. (47) 67157148372÷90009. (49) 7428927415293÷8496427. (50) 60435674536845÷79094451. (51) 65358547823÷5578. (52) 3968901531620÷687637943. (53) Divide 152181255 by 3854, and explain the process. (54) Divide 143255 by 4093. Explain the operation, and shew that it is correct. (55) Divide 203534191 by 72, first by Long Division, and then by its factors 8 and 9; and shew that the results in both cases coincide GREATEST COMMON MEASURE. 47. A MEASURE of any given number is a number which will divide the given number exactly, i.e. without a remainder. Thus, 2 is a measure of 6, because 2 is contained 3 times exactly in 6. When one number is a measure of another, the former is said to measure the latter. 48. A MULTIPLE of any given number is a number which contains it an exact number of times. Thus 6 is a multiple of 2. 49. A COMMON MEASURE of two or more given numbers is a number which will divide each of the given numbers exactly: thus, 3 is a common measure of 18, 27, and 36. The GREATEST COMMON MEASURE of two or more given numbers, is the greatest number which will divide each of the given numbers exactly: thus, 9 is the greatest common measure of 18, 27, and 36. 50. If a number measure each of two others, it will also measure their sum, or difference; and also, any multiple of either of them. Thus, 3 being a common measure of 9 and 15, will measure their sum, their difference, and also any multiple of either 9 or 15. The sum of 9 and 15=9+15=24=3×8; therefore 3 measures their sum 24. The difference of 15 and 9=15-9=6=2×3; therefore 3 measures their difference 6. Again, 36 is a multiple of 9, and 36=3 × 12; therefore 3 measures this multiple of 9; and similarly any other multiple of 9. Again, 75 is a multiple of 15; and 75=3× 25; therefore 3 measures this multiple of 15; and similarly any other multiple of 15. 51. To find the greatest common measure of two numbers. RULE. Divide the greater number by the less; if there be a remainder, divide the first divisor by it; if there be still a remainder, divide the second divisor by this remainder, and so on; always dividing the last preceding divisor by the last remainder, till nothing remains. The last divisor will be the greatest common measure required. Ex. Required the greatest common measure of 475 and 589. 475) 589 (1 475 114) 475 (4 19) 114 (6 114 0 therefore 19 is the greatest common measure of 475 and 589. Reason for the above process. Any number which measures 589 and 475, also measures their difference, or 589-475, or 114, Art. (50), also measures any multiple of 114, and therefore 4 × 114, or 456, Art. (50); and any number which measures 456 and 475, also measures their difference, or 475-456, or 19; and no number greater than 19 can measure the original numbers 589 and 475; for it has just been shewn that any number which measures them must also measure 19. Again, 19 itself will measure 589 and 475. For 19 measures 114 (since 114-6 × 19); 19 measures 4 × 114, or 456, Art. (50); therefore therefore 19 measures 456+19, or 475, Art. (50); therefore since 19 measures them both, and no number greater than 19 can measure them both, 19 is their greatest common measure. |