Factors and Multiples 1. What is the product of 8×12? 8 and 12 are called the factors of the product 96. The word factor means maker. Two or more numbers, which multiplied together make a number, are called factors of that number. 2. What is the product of 2×3×5? What is the product of 6X5? Name other pairs of factors of 30. The factors 2, 3 and 5 are called prime factors of 30. The factor 6 is called a composite factor because it may be considered as composed of two other factors, 2 and 3. A prime number is a number which has no exact divisor except itself and one. 2, 3, 5 and 23 are examples of prime numbers. 3. Write all the prime numbers between 1 and 50. A composite number is a number which has other exact divisors, except itself and one. 8, 15 and 27 are examples of composite numbers. 4. Write all the composite numbers between 1 and 50. Exercise 1 Give groups of either prime or composite factors of the following numbers: Practice in factoring numbers not only provides excellent training in division, but is also a valuable preparation for the following chapter on fractions. A quick recognition of the factors of a number aids in reducing a fraction to lower terms, in reducing fractions to a common denominator and in cancellation of common factors in multiplication of fractions. Divisibility of Numbers It is of great assistance in factoring numbers to be able to tell whether a number is exactly divisible by another number. There are certain short methods for finding whether a number is exactly divisible by certain other numbers. Exercise 2. Numbers Divisible by 2 1. Write the numbers from 1 to 40. 2. Divide each number by 2 to see which contains 2 exactly. A number which is exactly divisible by 2 is called an even number. A number which is not exactly divisible by 2 is called an odd number. 2, 4, 6 and 8 are even numbers and 1, 3, 5, 7 and 9 are odd numbers. 3. Study the numbers you have divided and give in your own words a short way of telling whether a number is exactly divisible by 2. Without pencil, tell which of the following numbers are exactly divisible by 2: A summary of all the tests of divisibility is given on pages 46 and 47. This summary is given as a check upon the statements formulated by the pupils. Exercise 3. Numbers Divisible by 4 1. See if the numbers from 1 to 40 which are divisible by 2 are also divisible by 4. 2. See if numbers which end in a digit (figure) that is divisible by 4 are always divisible by 4. Try such numbers as 244, 314, 128, 518 and 538. 3. See if numbers are divisible by 4 in which the numbers expressed by the last two digits (figures) are always divisible by 4. Try such numbers as 164, 224, 536, 784, 616 and 896. 4. See if numbers are divisible by 4 in which the numbers expressed by the last two digits are not divisible by 4. Try such numbers as 314, 561, 826, 935 and 722. 5. Tell in your own words, then, how we can tell which numbers are exactly divisible by 4. Without pencil, tell which of the following numbers are exactly divisible by 4: The term divisible in this chapter means exactly divisible. Exercise 4. Numbers Divisible by 8 1. See if numbers are divisible by 8 in which the number expressed by the last three digits of each are divisible by 8. Try such numbers as 4256, 9840, 12928 and 35168. 2. See if numbers are divisible by 8 in which the numbers expressed by the last three digits of each are not divisible by 8. Try such numbers as 3125, 7366, 11274 and 42612. 3. Give a short method, then, for telling when numbers are exactly divisible by 8. Without pencil, tell which of the following are divisible by 8: 1. Write all of the numbers from 3 to 144 which contain 3 an integral or whole number of times. 2. Find the sums of the digits in each number. Is each of these sums of the digits exactly divisible by 3? 3. Try other numbers which are not exactly divisible by 3. Are the sums of their digits divisible by 3? 4. What, then, is a short method of finding whether a number is exactly divisible by 3? Without pencil, tell which of the following numbers are divisible by 3: Pupils should use short division as a check upon their work in these exercises. This will give added facility in the process of short division. Exercise 6. Numbers Divisible by 6 1. See if the even numbers in Exercise 5 which are divisible by 3 are also exactly divisible by 6. 2. See if the odd numbers in Exercise 5 which are divisible by 3 are also exactly divisible by 6. 3. Find a short method of finding whether a number is divisible by 6. Which of the following numbers are divisible by 6? Exercise 7. Numbers Divisible by 9 The test for divisibility by 9 is very much like that for 3. 1. Write the numbers from 18 to 144 which you know contain 9 an integral number of times. 2. Find the sum of the digits in each number as in numbers divisible by 3. If you remember the test for 3, you can easily see the test for 9. 3. Tell in your own words the test of divisibility for 9. Without pencil, tell which of the following are divisible 21. By adding the digits as you write, make 10 numbers exactly divisible by 9. |