Question

Using the divisibility test, determine which of the following numbers are divisible by 2, 3, 4, 5, 6, 8, 9, 10, 11. Tabulate the data.
990; 1586; 275; 6686; 639210; 429714; 2856, 3060; 406839

Answer 1

Hint: To begin with we will define what is divisibility test and then recall the divisibility test of 2, 3, 4, 6, 8, 9, 10, and 11. Then we will take each example and find whether they are divisible by the given numbers or not. In the end, we will tabulate the data.

Complete step-by-step solution:
A divisibility test is a set of conditions, which defines whether a given number can be divided by a specific number or not.
Now, we shall recall the divisibility test of the given numbers:
2: If the digit at the unit’s place is an even, the whole number is divisible by 2.
3: If the sum of the digits is a number divisible by 3, then the number is divisible by 3.
4: If the number formed by the hundred’s and unit’s place of the number is divisible by 4, then the whole number is divisible by 4.
5: If the digit at the unit's place of the number is 5 or 0, then the number is divisible by 5.
6: If the number is divisible by 2 and 3, it is divisible by 6.
8: If the last 3 digits of a number are divisible by 8, then the number is divisible by 8. If the number is not divisible by 4, it cannot be divisible by 8.
9: If the sum of all the digits is divisible by 9, then the number is divisible by 9.
10: If the number has 0 in the unit’s place, then the number is divisible by 10.
11: Subtract the last digit from a number made by the other digits. If that number is divisible by 11 then the original number is, too.
The first number given to us is 990.
As we can see, it is divisible by 2 as the last digit is even. 9 + 9 + 0 = 18. It is divisible by 3 as well as 9. Thus, it will also be divisible by 6. 90 is not divisible by 4, hence 990 is not divisible by 4. It is not divisible by 8. It is divisible by 5 as well as 10, as units place is 0. 99 – 0 = 99. It is divisible by 11, hence the number is divisible by 11 too.
The second number is in 1586.
The unit’s place is 6, so it is divisible by 2. The sum of the digits is 20, which is not divisible 3 nor 9. So, it is not divisible by 6 too. 86 is not divisible by 4, so 1586 is also not divisible. If the number is not divisible by 4, it will not be divisible by 8. The last digit is neither 5, nor 0, so it neither divisible by 5, nor 10. 158 – 6 = 152, which is not divisible by 11, hence the number is not divisible by 11.
In a similar way, we will test for all the remaining numbers. After testing all the numbers, we will tabulate the data. The table is as follows:

	2	3	4	5	6	8	9	10	11
990	Yes	Yes	No	Yes	Yes	No	Yes	Yes	Yes
1586	Yes	No	No	No	No	No	No	No	No
275	No	No	No	Yes	No	No	No	No	Yes
6686	Yes	No	No	No	No	No	No	No	No
639210	Yes	Yes	No	Yes	Yes	No	No	Yes	Yes
429714	Yes	Yes	No	No	Yes	No	Yes	No	No
2856	Yes	Yes	Yes	No	Yes	Yes	No	No	No
3060	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	No
406839	No	Yes	No	No	No	No	No	No	No

Note: It is to be noted that a number is divisible by 9 only if it is divisible by 3 too. Vice-versa is not true. Similar is the case with 4 and 8, 5, and 10. An even number may be divisible by an even number or odd number, but no odd number is divisible by any even number.