Extension Divisibility Rules jamiedrfrostmaths com Dr Frost Maths

Extension : : Divisibility Rules jamie@drfrostmaths. com @Dr. Frost. Maths Objectives: Know the divisibility rules for 2 to 11, and know how to decompose into multiple divisibility rules for larger numbers. Last modified: 21 st January 2020

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Divisibility Rules How can we tell if a number is divisible by: ! 2 3 4 5 6 ? Digits add up to multiple of 3. e. g: 1692: 1+6+9+2 = 18 ? Last two digits together are divisible by 4. e. g. 143328 ? Last digit is 0 or 5. ? Number is divisible by 2 and 3 (so use tests for 2 and 3). ? Last digit is even. 7 Double the last digit and subtract it from the remaining number, and see if the result is divisible by 7. e. g: 2464 246 – 8 = 238 23 – 16 = 7. 8 Last three digits divisible by 8. 9 Digits add up to multiple of 9. 10 Last digit 0. 11 When you sum odd-positioned digits and subtract even-positioned digits, the result is divisible by 11. ? ? ? e. g. 47949: (4 + 9) – (7 + 4) = 22 – 11 = 11, which is divisible by 11. 12 Number divisible by 3 and by 4. ?

Quickfire Divisibility 4 6 7 ? ? ? 726 168 ? ? ? 9196 ? ? ? 252 ? ? ? ? ? ? 1001 ? ? ? 91 216 ? ? ? 87912 ? ? ? 9 11 ? ? ? ?

Quickfire Mental Primes Apart from the obvious instant checks (divisibility by 2, 5), we usually only have to mentally check 3, 7 and 11 to have a good ‘guess’ that a number is prime. 3 91 101 234567 131 781 751 221 7 ? 11 Is it prime? No ? Yes ? No ? ? ? ? Yes No Yes ? ? Advanced: 13 trick is “Quadruple last digit and add to remaining number. Is result divisible by 13?

Test Your Understanding Easier One: [JMO 1997 A 5] Precisely, one of the numbers 234, 23456, 2345678, 23456789 is a prime number. Which one must it be? Solution: 23456789? Harder One: [JMC 2012 Q 23] Peter wrote a list of all the numbers that could be produced by changing one digit of the number 200. How many of the numbers on Peter’s list are prime? A 0 B 1 C 2 D 3 E 4 Solution: A ?

Breaking Down Divisibility Rules Are these statements true or false? If we want to show that a number is divisible by 15, we can show it is divisible by 3 and 5. False True If we want to show that a number is divisible by 24, we can show it is divisible by 6 and 4. False True The problem is that 12 is divisible by 6 and 4, but it is not divisible by 24! We need to pick two numbers which are coprime, i. e. do not share any factors. How can we therefore test if a number is divisible by 24?

Quickfire What divisibility rules would we use if we wanted to test divisibility by: 18 45 36 40 2 and 9 rules 5 and 9 4 and 9 5 and 8 ? ? An ‘easy’ Year 10 Maths Olympiad problem: Find the smallest positive integer which consists only of 0 s and 1 s, and which is divisible by 12. • Since in must be divisible by 4, the only possibility for the last two digits is 00. ? • It must have at least three 1 s to be divisible by 3 (as we can’t have zero 0 s). • Therefore 11100 is the answer.

Exercise 1 (on provided worksheet) [JMC 2011 Q 2] How many of the integers 123, 234, 345, 456, 567 are multiples of 3? 4 [ ] 1 [ ] 2 [ ] 3 [ ] 4 [ ] 5 Answer: 5 2 3 [ ] 2005 [ ] 2007 [ ] 2009 [ ] 2011 [ ] 2013 ? [JMC 2004 Q 2] Which of the following numbers is exactly divisible by 7? [ ] 104 [ ] 106 [ ] 108 [ ] 110 [ ] 112 Answer: 112 ? Solution: 2011 5 ? [JMC 2016 Q 11] Which of the following statements is false? [ ] 12 is a multiple of 2 [ ] 123 is a multiple of 3 [ ] 1234 is a multiple of 4 [ ] 12 345 is a multiple of 5 [ ] 123 456 is a multiple of 6 Answer: 1234 ? [JMC 2003 Q 13] This is a prime year, since 2003 is a prime number. In the next ten years there is just one prime year. Which is it? [JMC 2005 Q 15] There are six different threedigit numbers, each of which contains all the digits 1, 3 and 5. How many of these three-digit numbers are prime? [ ] 0 [ ] 1 [ ] 2 [ ] 3 [ ] 4 Solution: 0 ? 6 [JMC 1999 Q 17] The 8 -digit number 1234*678 is a multiple of 11. Which digit is represented by *? Solution: 9 ?

Exercise 7 (on provided worksheet) 10 ? 8 [JMC 2012 Q 23] Peter wrote a list of all the numbers that could be produced by changing one digit of the number 200. How many of the numbers on Peter’s list are prime? [ ] 0 [ ] 1 [ ] 2 [ ] 3 [ ] 4 Solution: 0 ? ? 9 11 [JMC 2015 Q 24] A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45? Solution: 9090 ? ?

Exercise (on provided worksheet) 12 [JMO 2011 B 1] Every digit of a given positive integer is either a 3 or a 4 with each occurring at least once. The integer is divisible by both 3 and 4. What is the smallest such integer? Solution: ? 3444 13 [SMC 2012 Q 6] What is the sum of the digits of the largest 4 -digit palindromic number which is divisible by 15? Palindromic numbers read the same backwards and forwards, e. g. 7227. Solution: 24 ? 14 ? 15 ?

Exercise (on provided worksheet) 16 18 ? ? 19 17 ? ?