I. Divide the greater of the two by the less. II. Make the remainder a divisor, and the divisor a dividend, and find another remainder. III. Proceed in this way until there is no remainder, and the last divisor is the greatest common measure required. 99. You may perhaps ask how the rule is to shew when the two numbers have no common measure. The fact is, that there are, strictly speaking, no such numbers, because all numbers are measured by 1; that is, contain an exact number of units, and therefore 1 is a common measure of every two numbers. If they have no other common measure, the last divisor will be 1, as in the following example, where the greatest common measure of 87 and 25 is found. 25)87(3 75 -- 12)25(2 24 -- 1)12(12 12 -- 0 EXERCISES. Numbers. g. c. m. 6197 9521 1 58363 2602 1 5547 147008443 1849 6281 326041 571 28915 31495 5 1509 300309 3 What are 36 × 36 + 2 × 36 × 72 + 72 × 72 and 36 × 36 × 36 + 72 × 72 × 72; and what is their greatest common measure?--_Answer_, 11664. 100. If two numbers be divisible by a third, and if the quotients be again divisible by a fourth, that third is not the greatest common measure. For example, 360 and 504 are both divisible by 4. The quotients are 90 and 126. Now 90 and 126 are both divisible by 9, the quotients of which division are 10 and 14. By (87), dividing a number by 4, and then dividing the quotient by 9, is the same thing as dividing the number itself by 4 × 9, or by 36. Then, since 36 is a common measure of 360 and 504, and is greater than 4, 4 is not the greatest common measure. Again, since 10 and 14 are both divisible by 2, 36 is not the greatest common measure. It therefore follows, that when two numbers are divided by their greatest common measure, the quotients have no common measure except 1 (99). Otherwise, the number which was called the greatest common measure in the last sentence is not so in reality. 101. To find the greatest common measure of three numbers, find the g. c. m. of the first and second, and of this and the third. For since all common divisors of the first and second are contained in their g. c. m., and no others, whatever is common to the first, second, and third, is common also to the third and the g. c. m. of the first and second, and no others. Similarly, to find the g. c. m. of four numbers, find the g. c. m. of the first, second, and third, and of that and the fourth.