The process here followed admits of the following illustration : It is required to find the greatest length which is contained an exact number of times in each of the straight lines AX and BY. Since this required length (which we may call L) is to measure both AX and BY, it must measure their difference, viz., C and as L measures C and BY, it must measure any number of times C and the difference between BY and this multiple of C; take from BY as many times C as possible, and suppose there is a remainder, D—, after taking it eight times. As before, L must measure both C and D, and suppose we find that D is contained exactly (say twice) in C, D will be the length required; for since D measures C, it measures 8 × C + D or BY; and since it measures BY, it also measures BY + C or AX, and is therefore a common measure of AX and BY; and it has been shewn that L must measure D; and as D is the greatest measure of itself, D is the G. C. M. required. Fourth method. On examining the process last given, we notice that the question of finding G. C. M. of two large numbers is reduced to that of finding two smaller numbers having the same G. C. M. If, therefore, one of the two given numbers has a prime factor not contained in the other, it may be thrown out at once. Again, a factor, prime or not, which is common to both, may be taken out and set aside for ultimate multiplication into the last divisor; and all this may be done at any stage of the process. This method will be found useful where the factors can be detected by inspection. Find G. C. M. of 976800 and 9990. 1st stage. Take out and preserve the common factor 10. 2nd stage. Reject 5 × 210 from the left, neither factor being common; take out and preserve the common factor 3. 3rd stage. Reject 2 × 2 × 28 from the left, and 3 x 39 from the right, as not common. 4th stage. Reject 11 from the left. Find G. C. M. of 250387 and 41041. In this example, we commenced as usual, but the first division revealed the common factor 41, the two quotients 101 and 1001 proving to be prime to each other. 41 is G. C. M. required. 16. If the G. C. M. of more than two numbers be required, we must first find that of any two of them; then the G. C. M. of this result and a third number; then of this second result and a fourth, and so on. Find G. C.M. of 4994, 7491, 9988, 12485, 16571. 227 is therefore the G. C. M. of all the numbers. NOTE.-If of a series of numbers there be one which measures each of the others, it is the G. C. M. of the series. G.C. M. 3 x 37=111 Ans. 111. (4) (5) (6) (7) 1521, 585, 4095, 3393, 10764, 4563. (8) Find the largest number of which the following are multiples: 833, 1785, 1309. (9) An exact number of shares, all at the same price, was bought with each of the following sums: £87. 6s. 3d., £134. 18s. 9d., £341. 6s. 3d. Find the highest possible price of each share. (10) Two distances of 901 and 1037 miles respectively are portioned off into equal daily journeys. Find the smallest number of days in which the journeys can be accomplished. (11) A court 6 yds., 2 ft., 7 in. long, and 5 yds., 2 ft., 5 in. broad, is to be paved with square tiles. Find the largest possible size of the tiles, and how many are required? 17. LEAST COMMON MULTIPLE. The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, &c.; the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, &c. We see that these two numbers have the multiples 12, 24, 36, &c., in common, while 3, 6, 9, 15, 18, &c., belong only to 3, and 4, 8, 16, 20, &c., only to 4. Learn by heart: The multiples that two or more numbers have in common are called their common multiples, and the least of these is called their LEAST COMMON MULTIPLE, which is indicated by the letters L. C. M. 18. Find L. C. M. of 6 and 7. Write out simultaneously the multiples of 6 and 7; 6, 12, 18, 24, 30, 36, 42; 7, 14, 21, 28, 35, 42; until we find a number (42) in one series which has already occurred in the other. This number 42 is the L. C. M. required. Other common multiples will be found further on in the series, but all of them multiples of 42. The smallest (integral) common measure of integers is 1. There cannot be a greatest common multiple. Two numbers must have a common multiple; for 2 × 3 or 6, must be a multiple of 2 and of 3; similarly, 17 × 19 or 323, must contain 17, 19 times, and 19, 17 times; and the question arises whether this product of two numbers is always not only a common multiple, but their least common multiple. L |