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TESTS FOR DIVISIBILITY

This article explains the meaning of division and multiples, providing tests for divisibility by numbers such as 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Learn how to determine if a number is divisible by these factors.

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TESTS FOR DIVISIBILITY

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  1. TESTS FOR DIVISIBILITY Ms Hudson

  2. Research the meaning of these words!? • Division/Divisible • Multiple/Multiples • Even Number • Sum • Alternate • Prime Number • Composite Number • Factors • Difference

  3. Division by 2 - • All multiples of 2, including 2 itself, are called even numbers. An easy way to determine an even number is to tinspect that last digit. If it is a 0, 2, 4, 6 or 8, then the number is divisible by 2. • Examples of multiples of 2 are 60000396, 498, 461357354, or 111300.

  4. Division by 3 - • To find multiples of 3, add the digits of the number and if the sum is divisible by 3 then the number is a multiple of 3. • For example, the sum of the digits of 672 is 6 + 7 + 2 = 15. Since 15 is divisible by 3, we can say that 672 is a multiple of 3.

  5. Division by 4 - • For multiples of 4, only the last two digits in the number are considered. We need only consider the last two digits since 100 and multiples of 100 are divisible by 4. • For example the number 576 is a multiple of 4 since 76 is divisible by 4. Examples of multiples of 4 are 432, 6302916, 3296 .

  6. Division by 5 - • If the last digit is 0 or 5 then the number is a multiple of 5. • Examples of multiples of 5 are 3152905, 6690, 21375.

  7. Division by 6 - • To find the multiples of 6, we combine the tests for finding multiples of 2 and 3 because 6 = 2 x 3. This means that multiples of 6 must be even and the sum of the digits must be a multiple of 3. • Consider 312576. First, the number is even. Second, the digits add to 24 (3 + 1 + 2 + 5 + 7 + 6 = 24). Therefore 312576 is a multiple of 6!

  8. Division by 7 - • Unfortunately there is no easy test for the divisibility of 7.

  9. Division by 8 - • Since 1000 and multiples of 1000 are divisible by 8, we need to only consider the last three digits of a number to discover if it is a multiple of 8. • For example, 236 739 472 is a multiple of 8 since 472 is divisible by 8. Other example of multiples of 8 are 23528 and 40379232.

  10. Division by 9 - • The test for divisibility by 9 is similar to the test for divisibility by 3. Add the digits and if the sum is a multiple of 9 then the number is a multiple of 9. • The number 3645 is a multiple of 9 since 3 + 6 + 4 + 5 = 18, which is a multiple of 9.

  11. Division by 10 - • If the last digit of the number is 0, then the number is a multiple of 10…. • Examples of multiples of 10 are 670, 2000, 4619270. That should have been the easiest division rule of the lot!

  12. Division by 11 - • For multiples of 11, add alternate digits of the number, then add the remaining digits and if the difference between the two sums is obtained is 0, or a multiple of 11, then the number is a multiple of 11. • For example, consider the number 7968323. We see that 7 + 6 + 3 + 3 = 19 and 9 + 8 + 2 = 19, and the difference between these two sums is 0, hence 7968323 is a multiple of 11.

  13. Division by 12 - • The tests for divisibility by 12 requires the combined use of the tests for divisibility by 3 and 4. Any number divisible by 3 and 4, must be divisible by 12 because 3 x 4 = 12.

  14. The same rules apply! • In the same way, tests for divisibility by other composite numbers can be developed as combinations of these tests of the factors of the number; • For multiples of 15, test for divisibility of 3 and 5 • For multiples of 18, test for divisibility of 2 and 9.

  15. Check your knowledge! • Q1 – A test for division by 1 was not given. Why? • Q2 – What is another name for a multiple of 2? • Q3 – Explain why 2412 is divisible by 3? • Q4 – Explain why 237 is not divisible by 5? • Q5 – 237 is a multiple of which two numbers? • Q6 – Which digits are inspected for multiples of 4? • Q7 – Show that 2241 is not a prime number (i.e show that 2241 is divisible by some other number besides 1 and 2241). • Q8 – A certain number is a multiple of 3 and 7. Of what other number, besides 1, must it also be a multiple? • Q9 – Which do you think is the hardest test and why? • Q10 – Write a sentence to explain how you would test a number for divisibility by 30?

  16. Whole Numbers

  17. Divisibility Tests A number is divisible by Rule If it is even (ends in 0, 2, 4, 6 or 8) 2 3 If the sum of the digits is divisible by 3 4 If the number formed by the last 2 digits is divisible by 4 5 If it ends in 0 or 5 If it is divisible by 2 and 3 6 8 If the number formed by the last 3 digits is divisible by 8 9 If the sum of the digits is divisible by 9 10 If it ends in 0 If it is divisible by 3 and 4 12

  18. Factors The factors of a number are all the numbers that divide exactly into it. e.g: The factors of 15 are {1, 3, 5, 15} Do you understand the difference between factors and multiples? Note: the factors of a number include 1 and itself.

  19. Multiples The multiples of a number are obtained by multiplying it by the natural (counting) numbers. example 1: The multiples of 10 are: 10, 20, 30, 40……… example 2:The multiples of 6 are: It’s just like listing out your times tables 6, 12, 18, 24, ……………

  20. Prime Numbers • A prime number has exactly two factors, itself and 1. • 2 is the smallest prime number,1 is not a prime number. • Prime numbers are:{2, 3, 5, 7, 11, ………..} I’m prime NZ beef A Composite number is a natural number which has more than 2 factors. eg 9, 12 etc

  21. Factor trees & Product of Primes 84 • Start with ANY two numbers that multiply to give 84 E.g. Write 84 as a product of prime factors 42 2 21 We circle the prime numbers at the end of each branch. 2 7 3 Don’t forget to write the answer! 84 = 2 x 2 x 3 x 7 = 22 x 3 x 7

  22. Highest Common Factor (HCF) example: Find the HCF of 24 and 30 3, 8, 6, 4, The factors of 24 are: Method 1: 2, 12, 1, 24 5, 6, 15, The factors of 30 are: 2, 10, 3, 30 1, 2 24 30 Method 2: 12 15 3 4 5 The HCF of 24 and 30 is 6 HCF = 2 x 3 = 6

  23. Lowest Common Multiple(LCM) example: Find the LCM of 12 and 30 Multiples of 12 are: 12, 24, 36, 48, 60, 72........ Method 1: Multiples of 30 are: 30, 60, 90, 120…….. 2 12 30 Method 2: The LCM of 12 and 30 is 60 3 6 15 LCM = 2 x 3 x 2 x 5 = 60 2 5

  24. Index Notation index/power/exponent 4 x 4 x 4 x 4 x 4 = 45 4 5 is read as “4 to the power of 5” Two special names are squared, e.g. 4 squared = 4 x 4 = 4 2 cubed, e.g. 2 cubed = 2 x 2 x 2 = 2 3 base

  25. ( (-) 3 ) x 2 = Powers on a calculator Always enter negatives with brackets. (-3)2 = Calculators generally have 1. For x squared 2. For x cubed 3. For other powers To calculate 4 9 we would key in: and get the answer 262144 x 2 9 x 3   9 = 4

  26. Square Numbers & Square Roots If you multiply a number by itself you get a squarenumberie 1 x 1 = 1; 2 x 2 = 4; 3 x 3 = 9; 4 x 4 = 16 etc So 1, 4, 9, 16, 25, 36……. are square numbers The square root of 25 is 5, because 5 x 5 = 25 The symbol for square root is so That’s what you’ll put into your calculator It’s cool to be square

  27. Order of Operations Brackets BIMDAS Indices When both xand ÷ occur in a problem work from left to right. Multiplication Division Addition Subtraction When both + and - occur in a question work from left to right

  28. Working left to right subtraction comes first examples: 1) 8  6 + 2  4 2) 16  3 x 4  4 3) (16  3) x 4 = 52 4) 52 (2 x 32) + 4 ÷ 2 =

  29. It is very important to understand that it does make a difference if the order is not performed correctly!!!! Order of Operations

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