DIVISIBILITY OF NUMBERS

112. 1. Count by 2's to 20. Numbers that are exactly divisible by 2 are called even numbers. All numbers ending in 0, 2, 4, 6, and 8 are even numbers. Name the even numbers from 40 to 60.

2. Numbers that are not exactly divisible by 2 are called odd numbers. Name the odd numbers from 20 to 40. 3. Which of the following are even numbers: 18, 27, 31, 46, 50, 65, 123, 2456?

4. Some numbers are not exactly divisible by any whole number except themselves and 1. Such numbers are called prime numbers. Write the prime numbers below 80. Numbers that are exactly divisible by whole numbers other than themselves and 1 are called composite numbers.

5. The factors of 15 are 3 and 5. These two numbers when multiplied together give 15. Some numbers have many factors. 2, 3, 4, 6, 8, and 12 are each a factor of 24. By what must each be multiplied to give 24 ?

A number which when multiplied by another number makes a given number is called a factor of the given number.

6. Name the factors of 6; of 12; of 16; of 30; of 36. Has a prime number factors?

7. The prime factors of a number are the prime numbers which when multiplied together make the number. The prime factors of 24 are 2, 2, 3, 2.

8. Name the prime factors of 18; of 20; of 36.

9. A factor is always an exact measure of a number. What numbers are exact measures of 21? of 30? of 16? of 12? of 48? Is 24 an exact measure of 48?

10. All the prime factors of a number may be found by dividing the number by one of its prime factors, and dividing each quotient in turn by one of its prime factors until the quotient is a prime number. The prime factors of 60 may be found by dividing 60 by 2, and dividing the quotient (30) by 2, and this quotient (15) by 3. The last quotient (5) is a prime number. The prime factors of 60 are the divisors, 2, 2, 3, and the last quotient, 5.

11. Find the prime factors of 48; of 72; of 80.

12. Which of the following are prime numbers: 27, 17, 39, 51, 29, 91, 53, 89, 77, 57 ?

13. Count by 5's to 35, beginning with 5. Numbers ending in 5 and in 0 are exactly divisible by 5.

14. What numbers are exactly divisible by 10? by 2? by 5?

15. Write a number the sum of whose digits is 3. Is the number exactly divisible by 3? Is 51 exactly divisible by 3?

16. Write a number the sum of whose digits is 6, 9, 12, or some other multiple of 3. Is the number exactly divisible by 3? Show by several illustrations that the following statement is correct:

A number is exactly divisible by 3 if the sum of its digits is exactly divisible by 3.

17. Which of the following are exactly divisible by 3: 54, 177, 81, 52, 819, 57, 69, 71, 213, 105, 86, 1612?

18. Write a number the sum of whose digits is divisible by 9.

A number is exactly divisible by 9 if the sum of its digits is exactly divisible by 9.

19. Which of the following are exactly divisible by 9: 54, 504, 522, 711, 827, 218, 745, 891, 5375, 457?

20. Which of the following are exactly divisible by 2? by 3? by 5? by 9? by 10? 45, 61, 360, 207, 783, 53, 540, 117, 102, 107, 37, 97, 201, 855, 732, 380, 4320, 105?

21. A number that is exactly divisible by 9 is exactly divisible by 3.. Why"?

22. All multiples of even numbers are even numbers. Why?

23. The number denoted by the two right-hand figures of 216 is 16. Will 4 exactly divide 16? 200? 216?

24. Write a number of three or more places in which the number denoted by the two right-hand figures is some multiple of 4. Is the number exactly divisible by 4?

A number is exactly divisible by 4 if the number denoted by its two right-hand figures is exactly divisible by 4.

25. Which of the following are exactly divisible by 4: 112, 202, 420, 532, 514 ?

26. Centennial years that are divisible by 400 (1200, 1600, etc.) and other years divisible by 4 are leap years. Which of the following will be leap years: 1910, 1912, 1908, 1926, 1924, 1960, 1990, 2000, 2100 ?

27. Write 10 numbers that are exactly divisible by 3.

113. The three pairs of factors of 24 are 2 and 12, 3 and 8, 4 and 6. Name all the pairs of factors of each of the following, naming no factor larger than 20 :

PART II

FRACTIONS

114. 1. Draw a line 4 inches long.

Divide it into

two equal parts. What is each part called?

2. If 6 pupils are separated into three equal groups, what part of the pupils will each group contain?

3. Draw a line. Divide it into eight equal parts. What is each part called?

ᄒ ᄒ ᄒ D

to to to to to to to to to to 16 16 16 16 16

B represents

4. A represents the line undivided. the line as divided into two equal parts. What does C represent? D? E? What name is given to each part of the line in B? in C? in D? in E? In each case, how many of the parts does it take to equal the entire line?

5. The length of the line is represented in turn by 1, 2, 4, §, and 18. of the line === of the line. of the line == of the line.

16

of the line of the line =

16

of the line. of

6.

7.

What is the difference between of the line and of the line?of the line and of the line?

11. Show by dividing circles that of a circle is equal

to of a circle; that

that of a circle plus

of a circle is equal to § of a circle;

of a circle is equal to of a

circle; that of a circle is equal to 1 of a circle.

12. Show by dividing rectangles that, 2, 3, 1, and of a rectangle are equivalent parts.

13. Using objects, show that

as & of 12 objects; that

of 12 objects is the same of 12 objects is the same as

1. Line A is what part of line B? what part of line C? 2. If B is called 1, what is A? C? If C is called 1, what is A? B? If C is called 6, what is A? B?

3. If A is called 3, what is B? C? If A is called 4, what is B? C?

4. The ratio of line A to line B is 1. ratio of line A to line C? of line B to line

What is the

A? of line B

to line C? of line C to line B? of line C to line A?