| Week | Topic and action |
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| Week 42 | |
Divisibility Problems (2)
- One whole number is divisible by another if, after dividing, the remainder is zero
- Divisibility determined by the last digit.
- Divisible by 2: All even numbers are divisible by 2. i.e., all numbers ending in 0,2,4,6 or 8. Or any whole number is divisible by 2 if its last digit is divisible by 2, such as 12, 934, 2056, 3978, ...
- Divisible by 10: Any whole number ends in a 0 is divisible by 10, such as 20, 310, 2790, 91350,...
- Divisible by 5: Any whole number ends in a 0 or 5 is divisible by 5, such as 15, 70, 195, 790, ...
- Divisibility determined by the last two, or three digits.
- Divisible by 4: A number is divisible by 4 if the number formed by the last two digits is divisible by 4, such as 104, 2012, 30328, 50796,...
- Divisible by 8: A number is divisible by 8 if the number formed by the last three digits is divisible by 8, such as 1008,23032, 257072, 3957168,...
- Divisible by 16, 32, ...: A number is divisible by 16, 32,... if the number formed by the last four, five, ... digits is divisible by the corresponding number, 10016 is divisible by 16 and 300096 is divisible by 32.
- Divisibility determined by the sum of digits.
- Divisible by 3: A number is divisible by 3 if the sum of the digits is divisible by 3, such as 102, 321, 567, 1032,20301...
- Divisible by 9: A number is divisible by 9 if the sum of the digits is divisible by 9, such as 117, 225, 459, 2709, 30924,...
- Divisible by 6: A number is divisible by 6 if the sum of the digits is divisible by 3 and the last digit is divisible by 2, such as 132, 324, 768, 3072, ...
- Divisible by 11: A number is divisible by 11 if the difference between the sum of odd position digits and the sum of even position digits is divisible by 11, such as 1001, 2508, 4081, 909040, ...
- Divisible by 7:
First method
- 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.
For Example: 364 (Double the 4 to get 8. Subtract 8 from 36 to get 28 which is divisible by 7. Then 364 is divisible by 7).
Second method
- Take the last two digits in a number.
- Double the number from the rest of the digits and add the number from the above last two digits.
- Repeat the process for larger numbers.
For Example: 1302 (Double the 13 to get 26. Adding 2 to 26 to get 28, which is divisible by 7. Then 1302 is divisible by 7).
Third method
- Take the last three digits in a number.
- Get the difference between the number from the rest of the digits and the number from the above last three digits.
- Repeat the process for larger numbers.
For Example: 99113, 113 − 99 = 14, which is divisible by 7. Then 99113 is divisible by 7.
Fourth method
- Take the number and multiply each digit beginning on the right hand side by 1, 3, 2, 6, 4, 5. Repeat this sequence 1, 3, 2, 6, 4, 5 as necessary if the number is more than 6 digits.(*Note that the sequence 1, 3, 2, 6, 4, 5 comes from the remainders of 1, 10, 100, 1,000, 10,000, 100,000 divided by 7. 1,000,000 divided by 7 again will have remainder of 1.)
- Add the products.
- If the sum is divisible by 7, then the original number is divisible by 7.
For Example: Is 1050 divisible by 7?
1(6) + 0(2) + 5(3) + 0(1) = 21
21 is divisible by 7 and then 1050 is also divisible by 7.
While the first, second, and third method are very good in certain cases, the fourth method is in most case of a better method as it involves mainly simple digits arithmetic operations!)