Before proceeding to find the greatest common divisor of two numbers, we will show that any number which will divide two numbers exactly, will also divide their difference.

Suppose we have a common divisor of 636 and 276; this will also exactly divide 360, their difference. For, 636 is made up of the two parts 276 and 360, so that any number which will exactly divide 636, will also divide 276+360; if a divisor of 636 will at the same time divide one of its parts, 276, it will of necessity divide the other part, 360. Hence a common divisor of 636 and 276 is also a divisor of their difference, 360.

As the divisor which is common to 636 and 276, is also a divisor of 360, it must be a common divisor of 360 and 276, and consequently of 84, the difference between 360 and 276; and in general, when any two numbers have a common divisor, and we subtract any number of times the smaller number from the larger, the remainder will be exactly divisible by this common divisor.

What, now, is the greatest common divisor of 360 and

276.

The greatest divisor cannot exceed the less number, 276. But 276 will not divide the other number, 360, without a remainder, 84. Hence, the greatest divisor of 276 and 84 must be the greatest common divisor of 360 and 276. Again, dividing 276 by 84, we find 3, quotient, and 24, remainder. So the greatest common divisor of 84 and 24 is also the greatest common divisor of 276 and 84, and consequently of 360 and 276. Now, dividing 84 by 24, we find the quotient 3, and remainder 12. Finally, dividing 24. by 12, we find it is contained exactly twice; so that the greatest common divisor of 24 and 12 is 12: consequently, 12 is the greatest common divisor of 360 and 276. We will exhibit in one point of view the above.

OPERATION.

276)360(1

276

84)276(3
252

24)84(3

72

12)24(2

24

0

Hence, to find the greatest common divisor of two numbers, we deduce this

RULE.

Divide the greater number by the less, then the less number by the remainder; thus continue to divide the last divisor by the last remainder, until there is no remainder. The last divisor will be the greatest common divisor.

NOTE. When there are more than two numbers whose greatest common divisor is required, we must find the greatest common divisor of any two, and then find the greatest common divisor of this divisor thus found, and one of the remaining numbers; and thus continue until all the different numbers have been used.

What is the greatest common divisor of two or more numbers? Repent the rule for finding the greatest common divisor of two numbers. How do you proceed when there are more than two numbers?

EXAMPLES.

1. Find the greatest common divisor of 592 and 999

OPERATION.

592)999(1

592

407)592(1

407

185)407(2
370

37)185(5

185

0

From which we obtain 37 for the greatest common divisor of 592 and 999.

2. What is the greatest common divisor of 492, 744, and 906?

We first find the greatest common divisor of 492 and 744 by the following

OPERATION.

492)744(1

492

252)492(1

252

240)252(1
240

12)240(20
240

0

Therefore, the greatest common divisor of 492 and 744 is 12.

Again, proceeding with 12 and 906, we have the following

OPERATION.

12)906(75
900

6)12(2

12

We thus find 6 to be the greatest common divisor of 12 and 906, and consequently of the three numbers, 492, 744, and 906.

3. What is the greatest common divisor of 315 and 405? Ans. 45.

4. What is the greatest common divisor of 1825 and 2655? Ans. 5. 5. What is the greatest common divisor of 506 and 308? Ans. 22.

6. What is the greatest common divisor of 404 and 364 ? Ans. 4. 7. What is the greatest common divisor of 246, 372, and 522? 36. We are now prepared to proceed to the reduction of fractions.

Ans. 6.

We know (PROP. VI., ART. 34) that we can divide both numerator and denominator of a fraction by any number without altering its value. If we divide by the greatest common divisor, the resulting fraction will be in its lowest terms.

1

Therefore, to reduce a fraction to its lowest terms, we have this

RULE.

Divide both numerator and denominator by their greatest common divisor.

How do you reduce a fraction to its lowest terms?

EXAMPLES.

1. Reduce to its lowest terms.

We have already found (Ex. 1, ART. 35,) the greatest common divisor of 592 and 999 to be 37. Dividing both these terms by 37, we find 16 and 27 for quotients: hence,

, when reduced to its lowest terms, becomes 19. 2. Reduce to its lowest terms.

3. Reduce 38, f, f, to their lowest terms.

4. Reduce 315 to its lowest terms.
5. Reduce to its lowest terms.
6. Reduce 275 to its lowest terms.
7. Reduce to its lowest terms.
8. Reduce 4184 to its lowest terms.
9. Reduce 33523 to its lowest terms.

41223

10. Reduce 8768 to its lowest terms.

66

Ans.

We may frequently discover numbers, by inspection, which will divide both numerator and denominator without a remainder. When this is the case, we need not resort to the rule for obtaining the greatest common divisor, until we have divided by such numbers.