Divisibility Tests - Lesson Year 7 Maths

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1. Jasmine Vincent Divisibility Tests 7 th January 2025

2. Future Links Simplifying fractions Identifying properties of a number based on it’s prime factorisation Prerequisite Knowledge Identify multiples and factors Find all factor pairs of a number Identify common factors Identify common multiples Identify prime numbers Prime factorisation

3. Using the Dr Frost online platform Skills in this Lesson 163 Divisibility laws from 3 to 11 and dealing with larger divisors 163a Know if a number is divisible by 2, 3, 4, 5, 6, 8, 9 or 10. 163b Know if a number is divisible by 11. 163c Determine an unknown digit to make a number divisible by 3, 4, 6, 8 or 9. 163d Know if a number is divisible by a large number by combining divisibility rules. TEACHERS Generate a random worksheet involving skills in this PowerPoint (for printing or online task setting). STUDENTS Start an independent practice involving skills in this PowerPoint. drfrost.org/w/866 Clicking this box takes you to a single question practice for a subskill to allow you further Test Your Understanding opportunities. (e.g. drfrost.org/s/123a) drfrost.org/p/866 drfrost.org/s/123a

4. Contents For lessons covering many concepts, please click the below to navigate quickly to the relevant part of the lesson.

5. Prerequisite Check: Factors, Multiples and Primes 1 Write as a product of prime factors.     2 List the first multiples of     3 List all the factors of     4 5 Which of the numbers below are even?   and   6 What is the smallest prime number?   7 Is a prime or composite number?   so is divisible by and making it a composite number   8 Identify the HCF of and .     9 Identify the LCM of and .     ? ? ? ? ? ? ? ? 10 Identify the HCF and LCM of and .   HCF: LCM:   ? Is a prime number? Explain why or why not.   Yes, is a prime number because it has exactly factors; and itself.   ? Show all solutions

6. The Big Idea: Divisibility Rules Mrs Clark How do you know if a number is divisible by ?   It will end in a 0, 2, 4, 6 or 8 It will be an even number Both correct! How do you know if a number is divisible by ?   It will end in a …   … or a !   Exactly! 163a drfrost.org/s/

7. The Big Idea: Divisibility Rules Mrs Clark Do you know any other divisibility rules already? Factor Rule Last digit is even or Last digit is or Factor Rule Logan Any number that ends in a is divisible by   Great! Let’s see if we can fill in the rest of the table Last digit is   163a drfrost.org/s/

8. is divisible by and   ? Quickfire Questions Is divisible by ?   Is divisible by ?   Is divisible by ?   Is divisible by ?   Is divisible by ?   Is divisible by and/or ?   Yes, because it ends in an even digit. No, because it doesn’t end in a or a .   Yes, because it ends in a .   No, because it doesn’t end in a .   Yes, because it ends in an even digit. ? ? ? ? ?

9. The Big Idea: Divisibility Rules 163a drfrost.org/s/ Mrs Clark We can use divisibility laws for other numbers too! Some are more complicated than others… Nathan How am I meant to know if is divisible by something other than , or ?   Factor Rule Last digit is even or Last digit is or Last digit is Factor Rule

10. Do you notice anything special about the highlighted rows? The Big Idea: Divisibility Rules Mrs Clark For some divisibility tests you need to add the digits in the number together 163a drfrost.org/s/ Number Calculation Digit Sum Number Calculation Digit Sum Nathan They are all numbers in the times table; and their digit sum is also in the times table   A number is divisible by if the sum of its digits if divisible by   Is divisible by ?   The digit sum of is: is divisible by therefore is divisible by  

11. Number Calculation Digit Sum Number Calculation Digit Sum Do you notice anything special about these highlighted rows? The Big Idea: Divisibility Rules Mrs Clark For some divisibility tests you need to add the digits in the number together 163a drfrost.org/s/ Logan I’ve noticed that the numbers in the times table have a digit sum of   Will this always be true?

12. The Big Idea: Divisibility Rules 163a drfrost.org/s/ Mrs Clark Let’s look at the times table in more detail   Number Calculation Digit Sum Number Calculation Digit Sum Logan doesn’t fit the pattern; maybe the rule is the digits sum to either or ?   Nathan You should test it out on a bigger number you know is divisible by – like   Number Calculation Digit Sum Number Calculation Digit Sum is also in the times table; so the digit sum just needs to be divisible by as well!   A number is divisible by if the sum of its digits if divisible by  

13. is not divisible by or therefore is not divisible by or   ? is divisible by therefore is divisible by .   is divisible by therefore is divisible by   is not divisible by therefore is not divisible by .   is divisible by and therefore is divisible by both and .   is divisible by but not by , therefore is divisible by but not by   ? ? ? ? ? Quickfire Questions Is divisible by ?   Is divisible by ?   Is divisible by ?   Is divisible by , or both?   Is divisible by , or both?   Is divisible by or both?  

14. The Big Idea: Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is Factor Rule 163a drfrost.org/s/

15. Example: Checking divisibility by or   Factor Rule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is Factor Rule Is the number divisible by or ?   The divisibly rules we’ve seen so far either use the last digit or the digit sum. The last digit of is . This means that is divisible by , but not or . The digit sum is: is divisible by and therefore is divisible by and . is divisible by and .  

16. Test Your Understanding Factor Rule Last digit is even or The digit sum is a multiple of Last digit is or The digit sum is a multiple of Last digit is Factor Rule Divisible by                  Divisible by                  Tick the factor(s) that each number is divisible by ? ? ? ? ? ? ? Make a -digit number that matches each statement                     A multiple of   Largest possible multiple of   Multiple of and   ? ? ? ? ? 163a drfrost.org/s/ 1 2 a b c Show all solutions

17. Divisible by Divisible by Divisible by Divisible by Divisible by Divisible by Divisible by Divisible by

18. The Big Idea: Divisibility Rules Mrs Clark All these numbers are divisible by ; do you notice anything special about them?   163a drfrost.org/s/           It might help to look specifically at the last digits of each number   Hannah If we treat the last digits as a -digit number; they are all multiples of   A number is divisible by if the number formed by the last digits is divisible by   Can you explain why this works?

19. The Big Idea: Divisibility Rules 163a drfrost.org/s/ Mrs Clark Let’s look at the number         can be split into and   is divisible by – we can tell this because it ends in two zeros   Any number that ends in two zeros is divisible by ; as is divisible by ( then any number divisible by is also divisible by   Therefore; after we have separated the number into a multiple of and the remainder we only need to check if the remainder is divisible by   is divisible by therefore we know that is divisible by  

20. Example: Divisibility by   163a drfrost.org/s/       First; we split the into a multiple of and the rest of the number   We know is divisible by because it is a multiple of   is not divisible by therefore we know that is not divisible by   Is the number divisible by ?  

21. is divisible by is also divisible by Therefore is divisible by .   is divisible by is not divisible by Therefore is not divisible by .   is divisible by is divisible by Therefore is divisible by   ? ? ? Quickfire Questions Is divisible by ?   Is divisible by ?   Is divisible by ?  

22. The Big Idea: Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The digit sum is a multiple of Last digit is Factor Rule 163a drfrost.org/s/

23. The Big Idea: Divisibility Rules 163a drfrost.org/s/ True or False? All multiples of are even…   True or False? All multiples of are even…   True or False? All multiples of are even…   False True True False False True Only some multiples of are even     Viktor The multiples of that are even are also multiples of ; and all the even numbers are multiples of   A number is divisible by if it divisible by both and  

24. Example: Divisibility by   Is the number divisible by ?   ends in an even number therefore it is divisible by The digit sum is: is divisible by therefore is divisible by is divisible by both and therefore is divisible by   A number is divisible by if it divisible by both and   Is the number divisible by ?   ends in an even number therefore it is divisible by The digit sum is: is not divisible by therefore is not divisible by is divisible by but not by therefore is not divisible by  

25. ends in an even digit therefore is divisible by The digit sum is: is not divisible by therefore is not divisible by is not divisible by   ends in an even digit therefore is divisible by The digit sum is: is divisible by therefore is divisible by is divisible by   ends in an odd digit therefore is not divisible by We don’t need to check if the number is divisible by as well is not divisible by   ? ? ? Quickfire Questions Is divisible by ?   Is divisible by ?   Is divisible by ?  

26. The Big Idea: Divisibility Rules Mrs Clark This is what we’ve learned so far Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and The digit sum is a multiple of Last digit is Factor Rule 163a drfrost.org/s/ Logan which is why this works… maybe there are other divisibility rules we can find that work in this way?  

27. The Big Idea: Divisibility Rules Mrs Clark Can you remember the divisibility rule for ?   163a drfrost.org/s/ A number is divisible by if the number formed by the last digits of the number is divisible by   You can test if a number is divisible by in a similar way; this time we must look at the last digits to see if it is divisible by     Is divisible by ?   If is divisible by then we will know the whole number is divisible by   Viktor But how do we know if is divisible by ?   To quickly check if you can divide a number by – test to see if you can halve it times and get an integer answer     is divisible by , therefore is divisible by  

28. FYI: For Your Interest To quickly check if you can divide a number by – test to see if you can halve it times and get an integer answer   For larger numbers we can use a similar divisibility rule for as we used for .         can be split into and   is divisible by – we can tell this because it ends in three zeros   Any number that ends in three zeros is divisible by ; as is divisible by ( then any number divisible by is also divisible by   Therefore; after we have separated the number into a multiple of and the remainder we only need to check if the remainder is divisible by   is divisible by , we can tell by halving the number times (this works because   Therefore is divisible by  

29. is divisible by is not an integer therefore is not divisible by Therefore is not divisible by .   is divisible by is an integer therefore is divisible by Therefore is divisible by .   is divisible by is an integer therefore is divisible by Therefore is divisible by   ? ? ? Quickfire Questions Is divisible by ?   Is divisible by ?   Is divisible by ?  

30. Test Your Understanding Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and The last digits form a number divisible by The digit sum is a multiple of Last digit is Factor Rule 163a drfrost.org/s/ Divisible by                      Divisible by                      Tick the factor(s) that each number is divisible by ? ? ? ? ? ? Are the statements below always, sometimes or never true? If a number is a multiple of it is also a multiple of If a number is a multiple of it is also a multiple of   Always true Sometimes true ? ? 1 2 a b Show all solutions

31. Mrs Clark The divisibility rules for and are slightly less intuitive   A number is divisible by   if when adding and subtracting its digits in an alternative pattern it makes a number divisible by  .   Is divisible by ?   is divisible by therefore is divisible by   Is divisible by ?   drfrost.org/s/ 163b Is divisible by ?   is divisible by (and any number except ) therefore is divisible by   is not divisible by therefore is not divisible by   ?

32. Mrs Clark The divisibility rules for and are slightly less intuitive   A number is divisible by   if when you double the last digit and subtract it from the rest of the number, the result is divisible by  .   Is divisible by ?   Double the last digit: Number made by other digits: Subtract: is divisible by therefore is divisible by   Is divisible by ?   Double the last digit: Number made by other digits: Subtract: is divisible by therefore is divisible by   drfrost.org/s/ 163b ?

33. Double the last digit: Number made by other digits: Subtract: Now we need to check Double the last digit: Number made by other digits: Subtract: is not divisible by therefore, neither is therefore is not divisible by   ? A number is divisible by   if when you double the last digit and subtract it from the rest of the number, the result is divisible by  .   Is divisible by ?   Double the last digit: Number made by other digits: Subtract: Now we need to check using the same process Double the last digit: Number made by other digits: Subtract: is not divisible by therefore, neither is therefore is not divisible by   Is divisible by ?   drfrost.org/s/ 163b

34. The Big Idea: Divisibility Rules Factor Rule Last digit is even or The digit sum is a multiple of The last 2 digits form a number divisible by 4 Last digit is or The number is divisible by and Double the last digit, subtract from the remaining number, and see if the result if divisible by The last digits form a number divisible by The digit sum is a multiple of Last digit is Alternately add and subtract the digits, and see if the result is divisible by Factor Rule The last 2 digits form a number divisible by 4 Here’s a summary of all the divisibility rules you have learned:

35. Test Your Understanding Factor Rule Last digit is even or The digit sum is a multiple of The last digits form a number divisible by Last digit is or The number is divisible by and Double the last digit, subtract from the remaining number, and see if the result if divisible by 7 The last digits form a number divisible by The digit sum is a multiple of Last digit is Alternately add and subtract the digits, and see if the result is divisible by Factor Rule Double the last digit, subtract from the remaining number, and see if the result if divisible by 7 Divisible by                    Divisible by                    Tick the factor(s) that each number is divisible by ? ? ? ? ? ? 163a drfrost.org/s/ Show all solutions 163b

36. Apart from the obvious instant checks (divisibility by , ), we usually only need to mentally check and to have a good ‘guess’ that a number is prime.   Is it prime?  No Yes  No Yes  No Yes No! ( ) Is it prime?  No Yes  No Yes  No Yes N What is the largest prime factor we need to test before being certain a number is prime? We can use the square root of the number we are testing; for example so we would need to check up to 14. All composite numbers have a factor (other than 1) up to the square root.   ? ? The Big Idea: Prime Numbers ? ? ? ? ? ? ? ? ? ? ? ? ? 163b drfrost.org/s/

37. Find which possible digit(s) could go in the box to make    divisible by .   Recall the divisibility law for   A number is divisible by if it divisible by both and   For the number to be divisible by it would need to end in a or For the number to divisible by the digits needs to add up to a multiple of The digit sum so far is… This is already a multiple of therefore the digit in the box also needs to be a multiple of The missing digit could be or   Find which possible digit(s) could go in the box to make   divisible by .   Recall the divisibility law for   The two digit number  needs to be divisible by Consider the numbers in the times table around the number … … The missing digit could be or   A number is divisible by if the number formed by the last digits of the number is divisible by   drfrost.org/s/ 163c ?

38. For the number to be divisible by it needs to be divisible by and The number is already divisible by as it ends in a Digit sum: The digit sum is already a multiple of therefore the missing digit must also be divisible by : it could be or   For the number to be divisible by the last digits need to make a multiple of The last digits could be: or Out of this list only some are multiples of : the missing digit could be or   For the number to be divisible by the last digits needs to make a multiple of Thinking of the times table… … … The missing digit must be   Test Your Understanding Find which possible digit(s) could go in the box to make   divisible by .   Find which possible digit(s) could go in the box to make   divisible by .   ? Find which possible digit(s) could go in the box to make  divisible by .   For the number to be divisible by the digit sum needs to be a multiple of Digit sum: To make the digit sum a multiple of it would need to sum to : the missing digit must be   Find which possible digit(s) could go in the box to make   divisible by .   ? ? ? 163c drfrost.org/s/ 1 2 3 4

39. If we want to check if a number is divisible by , we can show it is divisible by and , they are coprime and have a product of   Are these statements true or false? If we want to show that a number is divisible by , we can show it is divisible by and .   If we want to show that a number is divisible by , we can show it is divisible by and .   The Big Idea: Combining Divisibility Rules False True True False           This is sometimes true; all of the multiples of are divisible by and . However, there are common multiples of and that are not divisible by . E.g. is divisible by and but not by .     We need to pick two numbers which are coprime, i.e. do not share any factors. Therefore, how should we test if a number is divisible by ?   ?

40. Quickfire Questions What divisibility rules would we use to test divisibility by:   and   and     and   and     and   and     and   and   163d drfrost.org/s/   and   and  

41. Find which possible digit(s) could go in the box to make   divisible by  .   For the number to be divisible by it must be divisible by and Divisible by Digit sum: Digit sum already a multiple of therefore missing digit can be or Divisible by Last digits need to be a multiple of : multiples of : , , , , Digits that fit both rules and can therefore be the missing digit: or   Find which possible digit(s) could go in the box to make   divisible by    For the number to be divisible by it must be divisible by and Divisible by Must end in a or Divisible by Digit sum: Digit sum already a multiple of therefore missing digit can be or The only digit that fits both rules and is therefore the missing digit:   drfrost.org/s/ 163d ? ? ? ?

42. Exercise (Available as a separate worksheet) Show all solutions 1 Tick the factor(s) that each number is divisible by Divisible by Prime                                     Divisible by Prime                                     ? ? ? ? ? ? [JMC 2011 Q2] How many of the integers are multiples of ? [   ]  [   ]  [   ]  [   ]  [   ]    Answer: (all numbers have a digit sum that is a multiple of )   2 [JMC 2004 Q2] Which of the following numbers is exactly divisible by ?   3 Answer:   ? ? ?

43. The number needs to be divisible by and ; the digit sum is and needs to be a multiple of Answer:   [JMC 2016 Q11] Which of the following statements is false? [   ]  is a multiple of [   ]  is a multiple of [   ]  is a multiple of [   ]  is a multiple of [   ]  is a multiple of   [JMC 2003 Q13] was a prime year, since is a prime number. In the following ten years there was just one prime year. Which was it? Hint: Use your divisibility rules! [   ]  [   ]  [   ]  [   ]  [   ]  Answer: is divisible by is divisible by and is divisible by is divisible by [JMC 1999 Q17] The -digit number is a multiple of . Which digit is represented by ? Answer: Therefore which creates a final sum of   Exercise (Available as a separate worksheet) Show all solutions 4 5 6 Answer: ( is not a multiple of )   7 8 ? ? Find which possible digit(s) could go in the box to make divisible by .   Current digit sum is ; this is already a multiple of Answer: or   ? Find which possible digit(s) could go in the box to make  divisible by    ? ?

44. [JMC 2000 Q17] The first and third digits of the five-digit number are the same. If the number is exactly divisible by , what is the sum of its five digits? Answer: Digit sum: must be an odd number between and , and the only odd multiple of in this interval is . [JMC 1997 Q20] A four-digit number was written on a piece of paper. The last two digits were then blotted out (as shown). If the complete number is exactly divisible by three, by four, and by five, what is the sum of the two missing digits? Answer: The missing digits are and   [JMC 2012 Q23] Peter wrote a list of all the numbers that could be produced by changing one digit of the number . How many of the numbers on Peter’s list are prime? [   ]  [   ]  [   ]  [   ]  [   ]  Answer: [Pink Kangaroo 2020 Q16] The digits from to are randomly arranged to make a -digit number. What is the probability that the resulting number is divisible by ? Answer:   Exercise (Available as a separate worksheet) Show all solutions 9 ? ? ? ? 10 11 12  

45. [SMC 2012 Q6] What is the sum of the digits of the largest -digit palindromic number which is divisible by ? Palindromic numbers read the same backwards and forwards, e.g. . Answer: For the number to be divisible by it eithers ends in a or a . It cannot end in a as the first digit cannot be . Therefore, the number is 5**5 The digit sum needs to be a multiple of , this is true if * or therefore the largest is when * which makes the digit sum   Exercise (Available as a separate worksheet) Show all solutions 13 The letters  ,   and   stand for non-zero digits. The integer ‘ ’ is a multiple of ; the integer ‘ ’ is a multiple of ; and the integer ‘ ’ is a multiple of . What is the integer ‘ ’? Answer: is a multiple of , therefore is a multiple of is a multiple of , the digit sum for would be the same therefore is a multiple of and Hence is a multiple of is a multiple of ; it must be a multiple of and as the digits are non-zero and is a multiple of The digit multiples of that start with a ( ) are: and Only has a as a multiple of Therefore and   N ? ?