Divisibility Rules
Recently I published a post on squaring numbers ending with five, but in this post, I'm going to take it a step further, and teach all the divisibility rules up to 13. Divisibility rules are the guidelines for working out if a larger number is divisible by your smaller one.
They never make an error.
Sadly, as there are infinite prime numbers, and therefore infinite divisibility rules, I can neither learn all of them, or teach you all of them. But chances are that you will ever need to know any above thirteen, so 'fire away!'.
1: All whole numbers are divisible by 1.
2, 4, 8, etc: To find out whether a number is divisible by 2^n, all you need to do is take the last n digits of the larger number is divisible by 2^n, or zero. e.g. 31242345840 is divisible by 8 (2^3), because the last 3 (n) digits, 840, is a multiple of 8. This is because any power of 2 (2^n) is a factor of the same power of ten (10^n). e.g. 16 (2^4) = 10000 (10^4) ÷ 625 (5^4). As you have just seen, you can go into even more detail with this rule by saying that a power of 2 (2^n) multiplied by the same power of five (5^n) equals the same power of ten (10^n).
3, 9 ,etc: If the digit sum is of the larger number is divisible by 3, than the larger number is divisible by 3. If the digit sum is also divisible by 9, than the number is divisible by 9 as well. e.g. 37281006 is divisible by both 9 and 3 because it has a digit sum of 27, which is divisible by both of these
6, 12, 15, 18, etc: To discover whether a number is divisible by six, you must use the divisibility rules for both numbers of a factor pair, which in this is 2 and 3. Same rules apply for 12, except obviously with a different factor pair. e.g. 7482 is divisible by six because it is divisible by both two and three.
5, 10: Any number ending with a 5 or 0 is a multiple of 5, but only numbers ending with a 0 are multiples of 10. e.g. 67389240987384340 is a multiple of 5 and 10 because it ends in 0.
11: If the difference between the odd digit sum and the even digit sum is a multiple of 11 or equal to 0, than the larger number is a multiple of 11. e.g. 1551667469 is a multiple of eleven because the difference between the odd digit sum (25) and the even digit sum (25) is equal to 0.
7, 13: If the difference between the last three digits of a number (n) is a multiple of 7, than the whole number is a multiple of 7. Same thing applies for 13. e.g. 23888111 is a multiple of 13 because 23888 - 111 = 23777 = 13 x 1829.
Hopefully these will come in handy.
Your's in numbers,
Lachlan
They never make an error.
Sadly, as there are infinite prime numbers, and therefore infinite divisibility rules, I can neither learn all of them, or teach you all of them. But chances are that you will ever need to know any above thirteen, so 'fire away!'.
1: All whole numbers are divisible by 1.
2, 4, 8, etc: To find out whether a number is divisible by 2^n, all you need to do is take the last n digits of the larger number is divisible by 2^n, or zero. e.g. 31242345840 is divisible by 8 (2^3), because the last 3 (n) digits, 840, is a multiple of 8. This is because any power of 2 (2^n) is a factor of the same power of ten (10^n). e.g. 16 (2^4) = 10000 (10^4) ÷ 625 (5^4). As you have just seen, you can go into even more detail with this rule by saying that a power of 2 (2^n) multiplied by the same power of five (5^n) equals the same power of ten (10^n).
3, 9 ,etc: If the digit sum is of the larger number is divisible by 3, than the larger number is divisible by 3. If the digit sum is also divisible by 9, than the number is divisible by 9 as well. e.g. 37281006 is divisible by both 9 and 3 because it has a digit sum of 27, which is divisible by both of these
6, 12, 15, 18, etc: To discover whether a number is divisible by six, you must use the divisibility rules for both numbers of a factor pair, which in this is 2 and 3. Same rules apply for 12, except obviously with a different factor pair. e.g. 7482 is divisible by six because it is divisible by both two and three.
5, 10: Any number ending with a 5 or 0 is a multiple of 5, but only numbers ending with a 0 are multiples of 10. e.g. 67389240987384340 is a multiple of 5 and 10 because it ends in 0.
11: If the difference between the odd digit sum and the even digit sum is a multiple of 11 or equal to 0, than the larger number is a multiple of 11. e.g. 1551667469 is a multiple of eleven because the difference between the odd digit sum (25) and the even digit sum (25) is equal to 0.
7, 13: If the difference between the last three digits of a number (n) is a multiple of 7, than the whole number is a multiple of 7. Same thing applies for 13. e.g. 23888111 is a multiple of 13 because 23888 - 111 = 23777 = 13 x 1829.
Hopefully these will come in handy.
Your's in numbers,
Lachlan
Divisibility rules test. Let’s see if you truly understand the above!
ReplyDeleteA number 5ab is repeated 99 times, so you get this: 5ab5ab5ab... 5ab5ab. This number is divisible by 91. What is 5ab?
You know I have no idea but is it 546
DeleteOops I did that really wrong
Deletewhat is it?
DeleteCorrection: For the add up the digits test for 3 and 9, they only work for 3 and 9 but not 27 or 81. Say you have a number abcd. You could expand that to 1000a + 100b + 10c + d.
ReplyDelete1000a + 100b + 10c + d is congruent to a + b + c + d mod 3.
1000a + 100b + 10c + d is also congruent to a + b + c + d mod 9, but not 27.
1000a + 100b + 10c + d is congruent to a + 19b + 10c + d mod 27. So, the rule for 27 is the first digit plus ten times the next digit plus nineteen times the next plus the next digit plus ten times the next and so on. 22383 is divisible by 27 because 3 + 10*8 + 19*3 + 2 + 10*2 is equal to 162 which is 27 * 6. Hope this helps as impractical as it is!