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The “g” Puzzle 37256

Have a classmate go to the board and multiply any 9

number of 4 or 5 digits by 9. You are to stand with

your back to the board so that you can not see his 335304 work. Tell him to draw a line through one of the

figures of the product. Have him read the other figures slowly so that you can add them. In the illustration the sum of 3, 3, 3, 0 and 4 is 13. We know, by the test for divisibility for 9, that the sum of all of the digits of a number is divisible by 9, if 9 is a factor of the number. The next higher number above 13 that is divisible by 9 is 18. 18—13=5. Therefore you can tell him that he crossed out a 5. This will hold for all digits except 0 and 9. Why can you not distinguish between crossing out a 0 and a 9 in the product?

Try this game with some one who has not played before and see if he can solve the puzzle.

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Exercise 8. Numbers Divisible by 5 1. Write the numbers from 1 to 60.

2. Check the ones which contain 5 an integral or whole number of times. Is there anything about their form which helps you to find them?

3. State in your own words the test for divisibility by 5. Without pencil, tell which of the following are divisible by 5:

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Summary-Tests for Divisibility 1. A number is divisible by 2 if it ends in 0 or any even digit. Such numbers are called even numbers.

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

3. A number is divisible by 4 if the number formed by the last two digits is divisible by 4. Only even numbers are divisible by 4.

4. A number is divisible by 5 if it ends in either 0 or 5.

5. Any even number is divisible by 6 if the sum of its digits is divisible by 3.

6. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.

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

Exercise 9. “Buzz'— A Division Drill This game is played as follows: Form in a line. Count very rapidly from 1 to 100 by passing from one pupil to another. Suppose that 7 is the “buzz” word. When a number comes to a player that is exactly divisible by 7 or contains the figure 7, the player must say "buzz" instead of the number. When 100 is reached begin over again. If a player says 14, 71, or some other number where he should

Playing "Buzz” have responded “buzz,” the leader should direct him to take his seat. The leader should put a player out if he does not respond quickly by saying either a number or "buzz." Keep counting rapidly until only one player is left.

For example, the counting goes as follows: 1, 2, 3, 4, 5, 6, “buzz,” 8, 9, 10, 11, 12, 13, "buzz," 15, 16, "buzz," 18, 19, 20, "buzz,” 22, 23, 24, 25, 26, "buzz," "buzz,” 29, 30, etc.

This game can be varied by having 6, 8 or 9 for the "buzz” number.

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Exercise 10. Multiples A number is a multiple of another number if it is exactly divisible by that number. 8 and 16 are both multiples of 4; 8 being considered as 4 multiplied by 2 and 16 as 4 multiplied by 4.

The number 16 is a multiple of 8 and 2 as well as a multiple of 4.

1. Give all of the multiples of 4 from 4 to 64. State the other factor in each of these multiples of 4.

2. Give all the multiples of 3 from 3 to 36. State the other factor in each multiple.

3. Give the multiples of 5 from 5 to 80; of 6 from 6 to 96; of 7 from 7 to 112; of 8 from 8 to 128; of 9 from 9 to 144. Give the other factor in each multiple.

4. The number 36 is a multiple of seven different numbers besides itself and 1. Name these seven numbers.

Name the different numbers of which each of the following is a multiple: 1. 12 5. 28 9. 21 13. 42

17. 54

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Exercise 11. Common Multiples The number 16 is a multiple of 8. It is also a multiple of 4. 16, then, being a multiple of both 8 and 4, may be said to be a common multiple of 8 and 4.

A common multiple of two or more numbers is one that is exactly divisible by each of the numbers.

You will need to be able to recognize a number which is a common multiple of two or more numbers in your study of fractions.

Find, without pencil, the least number which is exactly divisible by each of the following groups of numbers: 1. 2,3 11. 12, 18 21. 2, 3, 4 31. 2, 4, 10 2. 3,4 12. 12, 16 22. 2, 4, 8 32. 6, 4, 8 3. 2,4 13. 8, 9 23. 2, 6, 8 33. 2, 3, 11 4. 4, 8 14. 9, 15 24. 3, 4, 5 34. 4, 6,9 5. 6,8 16. 7, 14 26. 5, 6, 2 35. 5, 6, 4 6. 5, 7 16. 14, 21 26. 8, 4, 12 36. 9, 12, 18 7. 3,5 17. 16, 32 27. 3, 5, 10 37. 12, 15, 30 8. 8, 12 18. 18, 24 28. 7, 14, 21 38. 8, 14, 28 9. 6,9 19. 16, 20 29. 8, 12, 16 39. 12, 18, 36 10. 6, 14 20. 6, 15 30. 4, 6, 3 40. 12, 24, 36

Since the least number which may be exactly divided by all of the numbers in any one of these groups must be a common multiple of these numbers it is usually called the least common multiple.

Exercise 12

Did you have any difficulty in finding the least common multiple in any of the groups in Exercise 11? If so, study this exercise to find an easier way.

1. What are the prime factors of 12? (See page 41 for prime factors.) 2) 12 Solution: Divide 12 by 2, giving 6 as a quotient. Divide 2) 6

6 by 2, giving 3 as a quotient. The prime factors of 12–

2X2X3. Use prime numbers (1, 2, 3, 5, etc.) in a similar 3

way in factoring other numbers.

2. What are the prime factors of 36?

3. How many times is 12 contained in 36? Are all of the prime factors of 12 also found in the prime factors of 36?

4. Find the prime factors of 24 and 72. 72 is exactly divisible by 24. Are all of the prime factors of 24 also found in the prime factors of 72?

6. Use other pairs of numbers in which one is exactly divisible by the other until you see clearly the following principle:

If one number contains another number a whole number of times, it also contains all of the factors of that number. 6. Find the least common multiple of 12, 24 and 36. The least common multiple of these numbers must contain each of these numbers a whole number of times, and hence must contain all of the factors of each number.

The prime factors of 12=2X2X3.
The prime factors of 24=2X2X2X3.
The prime factors of 36=2X2X3X3.

The least common multiple then must contain 2 at least 3 times as a factor in order to include the three twos found as factors in 24. It must also contain two threes in order to include the two threes found as factors in 36.

The least common multiple, then=2X2X2X3X3=72.

Find the least common multiple of the following numbers by factoring: 7. 12, 16, 32 10. 16, 24, 36

13. 6, 12, 30 8. . 9, 12, 24 11. 8, 32, 48

14. 8, 12, 36 9. 14, 8, 28 12. 12, 21, 42

16. 12, 16, 64 The treatment of the least common multiple has been much simplified in this text book. Its chief use is in connection with finding the least common denominator in fractions. Since most of the denominators in fractions in practical use are small, most of the least common denominators (or multiples) can be easily seen without the necessity of factoring. Have pupils use the factoring method only when they can not readily find the least common multiple without it.

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