Divisibility
Divisibility by 6:
A number is divisible by 6 when it is divisible by 2 and by 3 at the same time.
Example: 12, 42, 720,………….
The three numbers are divisible by 2 and by 3. This means they are also divisible by 6.
Divisibility by 7:
A number is divisible by 7 when separating the first digit on the right, multiplying it by 2, then subtracting this product from what remains on the left and so successively, it equals to zero or a multiple of 7.
Examples:
The number 21 is divisible by 7 because if I separate the digit on the right, which is 1, and I multiply it by 2, I get 2. When I separate the 1 from 21, the number 2 remains. If I now subtract this 2 from the one I got from multiplying 1 by 2, I get: 2-2 = 0. If the difference is zero or a multiple of 7, then the number is divisible by 7.
As you can see, knowing if a number is divisible by 7 or not is quite complicated. It is most of the times better to prove it by simply dividing the number in question by 7. However, let's see how we can know if a number is divisible by 7 without carrying out the division:
3.9 Is 252 divisible by 7?
Yes, because if I separate the digit on the right, 2 and I multiply it by 2; 2 x 2 = 4, then I subtract it from 25 (which is what remains when separating the last digit on the right) I get: 25 - 4 = 21 and we already know 21 is divisible by 7.
3.10 Is 231 divisible by 7? Why?
3.11 Is 315 divisible by 7? Why?
3.12 Is 483 divisible by 7? Why?
Answers:
3.10
231 is divisible by 7 because I multiply the last digit on the right, el 1: 2 x 1 = 2. I subtract it from the remaining digits (23): 23 – 2 = 21. I separate the digit on the right (1) from 21 and I multiply it by 2: 2 x 1= 2. I subtract this value from the remaining digit (2): 2 – 2 = 0. This means that 231 is divisible by 7.
3.11
315 is divisible by 7 because I multiply the last digit on the right, el 5: 2 x 5 = 10. I subtract it from the remaining digits (31): 31 – 10 = 21. I separate the digit on the right (1) from 21 and I multiply it by 2: 2 x 1= 2. I subtract this value from the remaining digit (2): 2 – 2 = 0. This means that 315 is divisible by 7.
3.12
483 is divisible by 7 because I multiply the last digit on the right, el 3: 2 x 3 = 6. I subtract it from the remaining digits (48): 48 – 6 = 42. I separate the digit on the right (2) from 42 and I multiply it by 2: 2 x 2= 4. I subtract this value from the remaining digit (4): 4 – 4 = 0. This means that 483 is divisible by 7.
Divisibility by 9:
A number is divisible by 9 when the sum of its digits is 9 or a multiple of 9.
Example:
729 is a multiple of 9 because the sum of the digits
that compose the number 729 is: 7 + 2 + 9 = 18
and 18 is a multiple of 9.
Divisibility by 11:
A number is divisible by 11 when the difference between the sum of the digits in the odd position and the sum of the digits in the even position, from right to left, is equal to zero, 11 or a multiple of 11.
Example:
The number 987657 is a multiple of 11 because if you add the digits that occupy an even position: 8+6+7 = 21, and you add the digits that occupy an odd position: 9+7+5 = 21.
If the difference of these two sums is zero, 11 or a multiple of 11, the number is divisible by 11.
3.13 Is 143 divisible by 11?
3.14 Is 5665 divisible by 11?
3.15 Is 476432 divisible by 11?
Answers:
3.13 Yes because 1+3 – 4 = 0
3.14 Yes because 5+6-6-5 = 0
3.15 Yes because 4+6+3 =13 and 7+4+2 = 13 I subtract both quantities and I get:
13 - 13 = 0
