Divisibility Rules Presented By: Mr. Yarow

Divided by 1 578 398 48 1903 All numbers are divisible by 1… Now You Try: Are These Numbers Divided by 1? 578 398 48 1903

Divided by 2 All numbers that end with 0 and with even numbers (2, 4, 6, 8, …) are divisible by 2. Now You Try! Are these numbers divisible by 2? 458 1279 540 231

Dividing by 3 Now You Try! Are these numbers divisible by 3? Add up the digits of the number. If that number is divisible by 3, then the original number is. If your sum is still a bigger number, continue to add the digits For example, take the number 783: Add up the digits: 7 + 8 + 3 = 18 If the number is divisible by 3, then the original number is Is 18 divisible by 3? YES! 18 3 = 6 … If your sum is till a big number, continue to add the digits: 1 + 8 = 9 Is 9 divisible by 3? YES… 9  3 = 3 Example: 381 (3+8+1=12, and 12÷3 = 4) Yes BUT 217 (2+1+7=10, and 10÷3 = 3 1/3) No Now You Try! Are these numbers divisible by 3? a) 639 b) 56 c) 86 d) 360

Dividing by 4 If the last two digits together are divisible by 4 Ex: 1312 is (12÷4=3) BUT 7019 is not Now You Try! Are these number divisible by 4? 548 261 56 920

Dividing by 5 554 6890 345 902 If the number ends in 0 or 5 Ex: 175 is BUT 809 is not Now You Try! Are these numbers divisible by 5? 554 6890 345 902

Dividing by 6 If the number is divisible by 2 and 3 114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes 308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No Now You Try! Are these numbers divisible by 6? 897 258 630 345

Dividing by 7 Take the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also. Ex: 672 (Double 2 is 4, 67- 4 = 63, and 63 ÷ 7 = 9) Yes {OR continue: double 3 = 6. 6 – 6 = 0) 905 (Double 5 is 10, 90 - 10 = 80, and 80 ÷ 7= 11 3/7) No Ex: Is 175 divisible by 7 Here the last digit is 5 , we double it , that is 10. Now 10 is subtracted from 17 17 - 10 = 7 As 7 is divisible by 7, so also 175 OR  This rule is called L-2M. What you do is to double the last digit of the number X and subtract it from X without its last digit. For instance, if the number X you are testing is345678, you would subtract 16 from 34567. Repeat this procedure until you get a number that you know for sure is or is not divisible by seven. Then the X's divisibility will be the same. Now You Try! Are these numbers divisible by 7? a) 578 b) 398 c) 48 d) 1903

Dividing by 8 If the last 3 digits are divisible by 8 Better  If the number is divisible by 2, then by 2 again, and then by 2 again Ex: 109816 (816 ÷ 8 = 102) Yes (easier divide by 2, then by 2, then by 2) 216302 (302 ÷ 8 = 37 3/4) No Now You Try! Are these numbers divisible by 8? 568 396 490 1903

Dividing by 9 Similar to dividing by 3. Add up digits. If that number is divisible by 9 then your number is divisible by 9 For example, take the number 924,561 Add up the digits of the number: 9 + 2 + 4 + 5 + 6 + 1 = 27 … if 27 is divisible by 9, so is the original number: 27 ÷ 9 = 3 If your sum is still a big number, continue to add the digits 2 + 7 = 9, and 9 ÷ 9 = 1 Ex. 1629 (1+6+2+9 = 18, and again, 1+8 = 9) Yes 2013 (2+0+1+3 = 6) No Now You Try! Are these numbers divisible by 9? 578 398 48 1903 490

Dividing by 10 If the number ends with a 0 578 398 1905 490 Now You Try! Are these numbers divisible by 10? 578 398 1905 490

Dividing by 11 Alternately add and subtract the digits from left to right. (You can think of the first digit as being 'added' to zero.) If the result (including 0) is divisible by 11, the number is also. Example: to see whether 365167484 is divisible by 11, start by subtracting: [0+]3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is divisible by 11. OR  If you sum every second digit and then subtract all other digits and the answer is: 0, or divisible by 11 OR  Make two sums of digits and subtract them. The first sum is the sum of the first, third, fifth, seventh, etc. digits and the other sum is the sum of the second, fourth, sixth, eighth, etc. digits. If, when you subtract the sums from each other, the difference is divisible by 11, then the number X is too Ex: 1364 ((3+4) - (1+6) = 0) Yes 3729 ((7+9) - (3+2) = 11) Yes 5176 ((5+7) - (2+1+6) = 3) No OR  To test whether a number is divisible by 11 or not , alternately add and subtract the digits from left to right . If the result ( including 0) is 0, then the number is also invisible * Ex: Let us take an example: 3267: 3 - 2 +  6 - 7= 0, So 3267 is divisible by 11 * Ex: 3960 is divisible by 11 because 3 + 6 and 9 + 0 both add up to 9, so the difference is 0, which is a multiple of 11 ( 0 • 11) Now You Try! Are these number divisible by 11? 578 398 48 1903 490

Dividing by 12 If the number is divisible by both 3 and 4, it is also divisible by 12. Ex. 648 (6+4+8=18 and 18÷3=6, also 48÷4=12) Yes 916 (9+1+6=16, 16÷3= 5 1/3) No

Dividing by 13 Delete the last digit from the number, then subtract 9 times the deleted digit from the remaining number. If what is left is divisible by 13, then so is the original number. OR  This rule is called L+4M. What you do is to quadruple the last digit of the number X and add it from X without its last digit. For instance, if the number X you are testing is345678, you would add 32 to 34567. Repeat this procedure until you get a number that you know for sure is or is not divisible by thirteen. Then the X's divisibility will be the same.

More Divisibility Rules - Dividing By: 14: If X is divisible by 7 and by 2, then X is divisible by 14 15: If X is divisible by 5 and by 3, then X is divisible by 15 16: If the last four digits are divisible by 16, then X is too 17: This rule is called L-5M. See rules for 7 and 13 on how to apply. 18: If X is divisible by 9 and by 2, then X is divisible by 18 19: This rule is called L+2M. See rules for 7 and 13 on how to apply. 20: If X is divisible by 5 and by 4, then X is divisible by 20 21: If X is divisible by 7 and by 3, then X is divisible by 21 22:  If X is divisible by 11 and by 2, then X is divisible by 22 24:  If X is divisible by 8 and by 3, then X is divisible by 24 25:  If the last two digits of X are divisible by 25, then X is too Higher: You can use multiple rules for multiple divisors...for instance, to check if a number is divisible by 57, check to see if it is divisible by 19 and 3, etc., since 57 = 19 x 3...

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Thank you for paying attention… Mr. Yarow