115. A PROBLEM is something to be done; or, it is a question which requires a solution. The solution of a problem consists of the operations necessary for finding the answer to the question. To solve a problem is to perform the operations for finding the answer.

116. PROBLEM 1. To resolve or separate a number into its prime factors:

RULE. Divide the given number by any prime number greater than one, that will divide it; divide the QUOTIENT by any prime number greater than one that will divide IT, and so on till the quotient is prime. The several divisors and last quotient will be the prime factors sought.

Ex. 1. What are the prime factors of 30? Ans. 2, 3, and 5.

OPERATION.

2)30

3)15

5

It is immaterial in what order the prime factors are taken, though it will usually be most convenient to take the smaller factors first.

Ans. 2, 2, 2, and 3.

Ans. 2, 2, 3, and 7.
Ans. 3, 5, 5, and 5.

2. What are the prime factors of 24? 3. Resolve 84 into its prime factors. 4. Resolve 375 into its prime factors. 5. What are the prime factors of 3465? 6. What are the prime factors of 19800? 7. What are the prime factors of 1440? 8. What are the prime factors of 3150? 9. What are the prime factors of 2310? 10. What are the prime factors of 1728? 11. What are the prime factors of 1800? 12. What are the prime factors of 2448? 13. What are the prime factors of 4824? 14. What are the prime factors of 3648? 15. What are the prime factors of 8696? 16. What are the prime factors of 7264?

17. What are the prime factors of 5075?

115. What is a Problem? The solution of a problem? What is it to solve a

problem? 116. Rule for finding the prime factors of a number?

117. If a number has composite factors, they may be found by multiplying together two or more of its prime factors; thus, the prime factors of 12 are 2, 2, and 3, and the composite factors of 12 are 2 × 2, 2 × 3, and 2 × 2 × 3, i. e. the composite factors of 12 are 4, 6, and 12.

GREATEST COMMON DIVISOR.

118. A COMMON DIVISOR of two or more numbers is any number that will divide each of them without remainder; thus 3 is a common divisor of 12, 18, and 30.

119. The GREATEST COMMON DIVISOR of two or more numbers is the greatest number that will divide each of them without remainder; thus, 6 is the greatest common divisor of 12, 18, and 30.

NOTE. A divisor of a number is often called a measure of the number, also an aliquot part of the number.

120. PROBLEM 2. To find the greatest common divisor of two or more numbers.

Ex. 1. What is the greatest common divisor of 18, 30, and 48? Ans. 2 X 3 6.

2. What is the greatest common divisor of 60, 72, 48, and 84?

119. Greatest Common Divisor? Other names for divisor?

in finding the greatest common divisor.
plies to other factors. Hence,

The same remark ap

RULE 1. Resolve each number into its prime factors, and the continued product of all the prime factors that are common to all the given numbers will be the common divisor sought.

3. What is the greatest common divisor of 24, 40, 64, 80, 96, 120, and 192? Ans. 2 X 2 X 28. 4. Find the greatest common divisor of 15, 45, 75, 105, 135, 150, and 300.

Ans. 15.

5. Find the greatest common divisor of 25, 45, and 70.

Ans. 5.

6. Find the greatest common divisor of 24, 36, and 64.

Ans. 4.

7. Find the greatest common divisor of 24, 48, 72, and 88. 8. Find the greatest common divisor of 45, 75, 90, 135, 150, and 180.

9. I have three rooms, the first 11ft. 3in. wide, the second 15ft. 9in. wide, and the third 18ft. wide; how wide is the widest carpeting which will exactly fit each room? How many breadths will be required to cover each room?

1st Ans. 27 inches.

121. When the given numbers are not readily resolved into their prime factors, their greatest common divisor may be more easily found by

RULE 2. Divide the greater of two numbers by the less, and, if there be a remainder, divide the divisor by the remainder, and continue dividing the last divisor by the last remainder until nothing remains; the last divisor is the greatest common divisor of the two numbers.

If more than two numbers are given, find the greatest divisor of two of them, then of this divisor and a third number, and so on until all the numbers have been taken; the last divisor will be the divisor sought.

120. Rule for finding the greatest common divisor of two or more numbers? 121. Second rule for finding greatest common divisor?

10. What is the greatest common divisor of 14 and 20?

Before explaining this operation, four principles may be stated, viz.:

(a) Every number is a divisor of itself, the quotient being one; thus, 3 is contained in 3 once; 7 in 7 once.

(b) If one number divides another, the 1st will divide any number of times the 2d; thus, since 3 divides 12, it will divide 5 times 12, or any number of times 12.

(c) If a number divides each of two numbers, it will divide their sum and also their difference; thus, since 6 is contained five times in 30, and twice in 12, it is contained 5 +2 = 7 times in 30 +12 42; and 5 12 2-3 times in 30 F = 18.

(d) Not only will the greatest common divisor of two numbers divide their difference, but unless one of the numbers is a divisor of the other, it will also divide what remains after one of the numbers has been taken from the other as many times as possible; thus, the greatest divisor of 6 and 22 will divide 22 3 × 6=4.

122. It may now be shown, 1st, that 2 is a common divisor of 14 and 20, and 2d, that it is their greatest common divisor.

First, 2 divides 6, .. (Art. 121, b) 2 divides 6 X 212, and (Art. 121, c) 2 divides 21214; again, since 2 divides 6 and 14 (Art. 121, c) it divides 6+14= 20; i. e. 2 divides both 14 and 20.

Second, The greatest divisor of 14 and 20 (Art. 121, c) must divide 20 14: 6, ..it cannot be greater than 6; again, the greatest divisor of 6 and 14 (Art. 121, d) must divide 14

121. First principle? Second? Third? Fourth? 122. Explain why 2 is a common divisor of 14 and 20. Why it is their greatest common divisor.

6 X 2 2,.. the greatest common divisor of 14 and 20 cannot exceed 2, and, as it has been previously shown that 2 is a divisor of 14 and 20, it is their greatest common divisor.

A similar explanation is applicable in all cases.

123. It will be seen that, in finding the common divisor of 14 and 20, we are led to find the divisor of 6 and 14, then of 2 and 6; i. e.. in any example we seek to find the measure of the remainder and divisor, then of the next remainder and divisor, and so on, until the greatest measure of the last remainder and the divisor which gave that remainder is found, and this measure will be the greatest common divisor of the two given numbers. Thus the question becomes more and more simple as each successive step in the operation is taken.

11. What is the greatest common divisor of 3432 and 4760?

The plan of the operation in Ex. 10 requires more space and more time than this in Ex. 11, though the principle and the reasoning are precisely the same in both.

In Ex. 11 we first divide 4760 by 3432, and obtain 1 for quotient and 1328 for remainder; then divide 3432 by 1328, obtaining 2 for quotient, and 776 for remainder; and so proceed, dividing the last divisor by the last remainder, as directed in Rule 2, until the remainder is 0. The last divisor, 8, is the

greatest common divisor of 3432 and 4760.

12. What is the greatest common divisor of 1430 and 3549? Ans. 13.

13. What is the greatest common divisor of 3640 and 5733 ? 14. What is the greatest common divisor of 1440 and 3696? 15. What is the greatest common divisor of 2520 and 6237?

123. Explain the operation in Ex. 11.