CHAPTER IX

FACTORS, MEASURES, MULTIPLES

158. Factors. Integral numbers whose product is a given number are called factors of that number.

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Thus, since 2 × 3 × 5 × 7 210, some of the factors of 210 are 2, 3, 5, and 7. We also see that 210 has other factors, like 6, 10, 14, and so on. Although every number has as factors itself and 1, we do not usually speak of these as factors.

159. Prime Number. A number that has no factors is called a prime number.

As stated in § 158, this means that the number itself and 1 are excluded. Thus 2, 3, 5, and 7 are prime numbers.

160. Prime Factor. A factor that is a prime number is called a prime factor.

Thus 2, 3, and 5 are the prime factors of 30. Although 6 is a factor of 30, it is not a prime factor.

161. Composite Number. A number that is not a prime number is called a composite number.

Thus 4, 15, 100, and 250 are composite numbers.

162. Exact Divisor. A number that divides another without a remainder is called an exact divisor of that number.

163. Exact Division. When we have an exact divisor, the case is said to be one of exact division.

Thus 15 is exactly divisible by 5. Usually we speak of an exact divisor simply as a divisor of a number. Then again we often speak of an exact divisor as a measure of a number, because with it we may exactly measure the number, just as with a yardstick we may measure the length of a carpet that is 10 yd. long.

164. Divisible Numbers. When we speak of one number as being divisible by another we mean exactly divisible, that is, divisible without a remainder.

165. Even Number. A number that is divisible by 2 is called an even number.

166. Odd Number. A number that is not divisible by 2 is called an odd number.

167. Divisibility by 2. A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.

Thus 36 must be divisible by 2 if 6 is; because 30, or any other number of tens, is divisible by 2, ten being 2 × 5.

Thus we may tell without dividing that 1,023,578 is divisible by 2.

168. Divisibility by 3. A number is divisible by 3 if the sum of the digits is divisible by 3.

We often use the word digits to mean the value of the symbols 0, 1, 2, and so on to 9, as well as the characters from 1 to 9.

Thus 54 is divisible by 3 since 5 + 4 is divisible by 3.

Likewise, 534 is divisible by 3 because 5+ 3 + 4, or 12, is divisible by 3.

169. Divisibility by 5. A number is divisible by 5 if it ends in 0 or 5.

For if a number ends in 0, it is tens, and 10 is divisible by 5. If it ends in 5, it is tens plus 5, and both are divisible by 5.

There is no easy test for divisibility by 7.

170. Finding Prime Factors. Find the prime factors of 420.

By § 167, 2 is a factor of 420 and of 210; by § 168, 3 is a factor of 105; by § 169, 5 is a factor of 35, the other factor being 7. Hence the prime factors of 420 are 2, 2, 3, 5, 7.

2)420 2)210 3)105

5)35

That is, divide as often as possible by the smallest prime factor, and then by the next larger, and so on until a prime quotient is reached. The several divisors and the last quotient are the prime factors.

171. Proving a Number Prime. Required to find if 239 is a prime number.

The last digit is not 0, 2, 4, 6, or 8, so 239 cannot contain the factor 2, and therefore cannot contain the factor 4, 6, 8, or any other multiple of 2. The sum of the digits does not contain the factor 3, and therefore 239 cannot contain the factor 3 or any multiple of 3. The last digit is not 0 or 5, and therefore 239 cannot contain the factor 5 or any multiple of 5. Therefore 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, cannot be factors of 239. By trial we find that 7, 11, 13, 17, are not factors. We need not try any prime number higher than 17, as the quotient when 17 is tried is less than 17, and we have tried all prime factors below 17. Hence 239 is prime.

Try the various prime numbers, beginning with 2, to see if they are factors. If no prime factor is found before the quotient becomes less than the divisor, the number is prime.

EXERCISE 94

1. Select the even numbers: 432, 640, 287, 1024, 3036, 8292, 4427, 8329, 4878.

2. Select the numbers divisible by 3: 424, 636, 292, 444, 1002, 3027, 1104, 8822, 6777.

3. Select the numbers divisible by 5: 830, 272, 637, 4225, 3010, 7200, 6305, 9270, 4005.

4. Select the prime numbers: 53, 61, 97, 111, 121, 143, 187, 1005, 1001, 1203, 2112, 1331, 1337.

172. Common Measure. A number that exactly divides each of two or more numbers is called a common measure of those numbers.

Thus 7 is a common measure of 28 and 42, since it exactly divides each of them, as a foot length exactly divides a 10-foot length and a 12-foot length that we are measuring. A common measure is also called a common divisor, or a common factor.

173. Greatest Common Measure. The greatest number that exactly divides each of two or more numbers is called the greatest common measure of those numbers.

Thus, while 7 is a common measure of 28 and 42, 14 is the greatest common measure. It is also called the greatest common divisor. The letters G.C.M. stand for the words Greatest Common Measure, and G.C.D. for Greatest Common Divisor.

174. Numbers Prime to Each Other. If two integers have no common measure, always excepting 1, they are said to be prime to each other.

Thus, 25 and 33 are prime to each other, although each has factors. 175. Finding the G.C.M. The greatest common measure of such numbers as we shall use in studying fractions is found easily by factoring.

For example, required the G.C.M. of 24, 30, and 36.
Finding the prime factors of each, we have the following:

The factor 2 occurs once in each of the numbers. The factor 3 occurs once in each of the numbers. No other factor occurs in each one of the numbers. Therefore the G.C.M. is 2 × 3, or 6.

This solution may be expressed as follows:

24 = 2 × 2 × 2 × 3; 30 = 2 × 3 × 5; 36 = 2 × 2 Therefore the G.C.M. is 2 × 3, or 6.

3 x 3.

Required the G.C.M. of 54, 81, and 108.

Here 3 occurs three times as a factor in each number, and there is no other common factor. Hence the G. C.M. is 3 × 3 × 3, or 27.

Hence, to find the G.C.M. of two or more numbers, Separate the numbers into prime factors and find the product of the prime factors common to the numbers.

If one of the given numbers is a multiple of one of the others, it may be canceled, since it must contain all the factors of that other. Thus the G.C.M. of 48, 54, and 108 is the same as that of 48 and 54.