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8. Suppose we wish to know whether the numbers 204 and 468 have a common factor, we proceed as follows: We decompose them into their prime factors, and thus obtain 204-22 × 3 × 17, and 468-22 × 32 × 13. Here we see that 22 × 3 is common to both the numbers 204 and 468.

Hence, to find the greatest factor which is common to two or more numbers, or, as generally expressed, to find the greatest common measure of two or more numbers, we have this

RULE.

Resolve the numbers into their prime factors, (by Rule under Art. 7.) Then select such of the primes as are common to all the numbers, multiply them together, and the product will be the greatest common measure.

EXAMPLES.

1. What is the greatest common measure of 1326, 3094, and 4420?

These numbers, when resolved into their prime factors, become

1326-2 × 3 × 13 × 17

3094-2 × 7× 13 × 17

4420-22 × 5 × 13 x 17

The factors which are common, are 2, 13, and 17; therefore, the greatest common measure is 2× 13× 17= 442.

2. What is the greatest common measure of 556, 672, and 840? Ans. 22-4.

3. What is the greatest common measure of 110, 140, and 680? Ans. 2×5=10. 4. What is the greatest common measure of 255, and Ans. They have none.

532 ?

5. What is the greatest common measure of 375, 408, and 922? Ans. They have none.

9. We may also find the greatest common measure of two numbers by the following

RULE.

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

NOTE. Where there is no common measure, the last divisor will be 1.

EXAMPLES.

1. What is the greatest common measure of 360 and 630?

Operation.

360)630(1
360

270)360(1
270

90)270(3

270

0

Hence, the greatest common measure is 90.

2. What is the greatest common measure of 922, and 408? Ans. 2.

3. What is the greatest common measure of 1825, and 2555?

Ans. 365.

4. What is the greatest common measure of 124, and 682 ? Ans. 62.

5. What is the greatest common measure of 296, and 407? Ans. 37.

6. What is the greatest common measure of 404, and 364 ? Ans. 4.

7. What is the greatest common measure of 506, and 308? Ans. 22.

8. What is the greatest common measure of 212, and 416? Ans. 4.

9. What is the greatest common measure of 74, and 84? Ans. 2.

10. Suppose we wish to know what is the least. number which will divide by 215 and 460; we proceed as follows: We decompose them into their prime factors, and thus obtain 215-5 x 43, 460-22 x 5 x 23. Hence, we see that 22 × 5 × 23 × 43=19780, is the least number which can be divided by 215 and 460.

Hence, to find the least number which will divide by two or more numbers, or, as generally expressed, to find the least common multiple, we have this

RULE.

Resolve the numbers into their prime factors, (by Rule under Art. 7); select all the different factors which occur, observing when the same factor has different powers, to take the highest power. The continued product of the factors thus selected will be the least common multiple.

EXAMPLES.

1. What is the least common multiple of 12, 16, and

24 ?

These numbers, resolved into their prime factors, give

12=2a × 3
16=24

24=23×3

4

Therefore, 24x3=48 is the least multiple required. 2. What is the least common multiple of 9, 12, 16, 20, and 35? Ans. 2a × 32 × 5 × 7=5040. 3. What is the least common multiple of 7, 13, 39, and 84 ? Ans. 22×3×7×13=1092. 4. What is the least common multiple of the nine digits? Ans. 23 × 32 × 5×7=2520. common multiple of 3, 5, 7, 12, Ans. 22 × 32 × 5×7=1260.

5. What is the least

15, 18, and 35?

6. What is the least common multiple of 100, 109, 463, and 900 ?

Ans. 22 × 32 × 52 × 109 × 463=45420300. 7. What is the least common multiple of 365, 910, 2217, and 2424 ?

23×3×5×7 × 13 × 73 × 101 × 739=

Ans.} 59499225240.

11. We may also find the least common multiple of two or more numbers by the following

RULE.*

Write the numbers in a horizontal line; divide them by any prime number which will divide two or more of them; place the quotients with the undivided terms for

* This rule is usually given as follows: "Write down the numbers in a line, and divide them by any number that will measure two or more o' them, and write the quotients and undivided numbers in a line beneath. Divide this line as before, and so on, until

a second horizontal line; proceed with this second line as with the first, and so continue, until there are no two terms which can be divided. The continued product of the divisors and the numbers in the last horizontal line will be the least common multiple.

EXAMPLES.

1. What is the least common multiple of 28, 35, 42, 77, and 70?

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Hence, 7×5×2×2×3×11=4620, is the multiple sought.

there are no two numbers that can be measured by the same divisor; then the continual product of all the divisors and numbers in the last line will be the least common multiple required.”

The above we have copied from Mr. Adams' Arithmetic. Nearly all our Arithmetics give, in substance, the same rule. We will now show, by an example, that this rule may give very different results, depending upon the divisors used, and of course the rule is in fault.

EXAMPLE.

What is the least common multiple of 12, 16, and 24?

We will work this example in three ways, as follows:

First Operation. 12 12, 16, 24 2 1, 16, 2 1, 8, 1

12×2×8=192.

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1,

2, 1. 1 2, 1 8×3×2×2=96. 4×3×2×2=48.

These operations, which are wrought strictly by this rule, give 192, 96, and 48, for the least multiple of 12, 16, and 24. Hence, the rule is wrong, and cannot be depended upon. The least common multiple of 12, 16, and 24, is 48, as may be found by either of our rules.

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