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Characteristic of number Last digit is even Number divisible by: 2 The sum of the digits is divisible by 3 3 The last two digits form a number divisible by 4 4 The last digit is 0 or 5 5 The number is divisible by both 2 and 3. 6 To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number. If you get an answer divisible by 7 (including zero), then the original number is divisible by seven. If you don't know the new number's divisibility, you can apply the rule again. Example: Check to see if 203 is divisible by 7. · · · 7 double the last digit: 3 x 2 = 6 subtract that from the rest of the number: 20 - 6 = 14. check to see if the difference is divisible by 7: 14 is divisible by 7, therefore 203 is also divisible by 7. The last three digits form a number divisible by 8 8 The sum of the digits is divisible by 9 9 The numeral ends in 0 10 The (sum of the odd numbered digits) - (sum of the even numbered digits) is divisible by 11. Example: 34871903 3+8+1+0=12 4+7+9+3=23 23-12=11 Is divisible by 11 The number is divisible by both 3 and 4. 11 12 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. 13 The last four digits form a number divisible by 16 16 TEST TIP: If a number is divisible by two different prime numbers, then it is divisible by the products of those two numbers. Since 36, is divisible by both 2 and 3, it is also divisible by 6. How can you tell whether a number is divisible by another number (leaving no remainder) without actually doing the division? Why do "divisibility rules" work? For examples and explanations, see Divisibility Rules. Divisibility by: 2 If the last digit is even, the number is divisible by 2. 3 If the sum of the digits is divisible by 3, the number is also. 4 If the last two digits form a number divisible by 4, the number is also. 5 If the last digit is a 5 or a 0, the number is divisible by 5. 6 If the number is divisible by both 3 and 2, it is also divisible by 6. 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. 8 If the last three digits form a number divisible by 8, then so is the whole number. 9 If the sum of the digits is divisible by 9, the number is also. 10 If the number ends in 0, it is divisible by 10. 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. 12 If the number is divisible by both 3 and 4, it is also divisible by 12. 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. [You may also want to look at a more complex method that can be extended to formulas for divisibility for prime numbers. Also, the idea of deleting the last digit and adding or subtracting a multiple of the digit from the remaining number can be generalized to test for divisibility by prime numbers up to 50 and beyond.] Divisibility Math Tricks to Learn the Facts (Divisibility) More and more in my teaching career, I see that we often are able to enhance student learning in mathematics with tricks. There are many tricks to teach children divisibility in mathematics. Some tricks that I used to use in my classroom are listed here. If you know of some that I may have missed, drop into the forum and let everyone know. I'll add them to this list as I see them. Dividing by 2 1. All even numbers are divisible by 2. E.g., all numbers ending in 0,2,4,6 or 8. Dividing by 3 1. 2. 3. Add up all the digits in the number. Find out what the sum is. If the sum is divisible by 3, so is the number For example: 12123 (1+2+1+2+3=9) 9 is divisible by 3, therefore 12123 is too! Dividing by 4 1. 2. 3. Are the last two digits in your number divisible by 4? If so, the number is too! For example: 358912 ends in 12 which is divisible by 4, thus so is 358912. Dividing by 5 1. Numbers ending in a 5 or a 0 are always divisible by 5. Dividing by 6 1. If the Number is divisible by 2 and 3 it is divisible by 6 also. Dividing by 7 (2 Tests) · · · · · · · · · · Take the last digit in a number. Double and subtract the last digit in your number from the rest of the digits. Repeat the process for larger numbers. Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7. NEXT TEST Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary Add the products. If the sum is divisible by 7 - so is your number. Example: Is 2016 divisible by 7? 6(1) + 1(3) + 0(2) + 2(6) = 21 21 is divisible by 7 and we can now say that 2016 is also divisible by 7. Dividing by 8 This one's not as easy, if the last 3 digits are divisible by 8, so is the entire number. 2. Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008. 1. Dividing by 9 1. 2. 3. Almost the same rule and dividing by 3. Add up all the digits in the number. Find out what the sum is. If the sum is divisible by 9, so is the number. For example: 43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too! Dividing by 10 1. If the number ends in a 0, it is divisible by 10.