Divisibility (is n divisible by 2, divisible by 3, divisible by 4, divisible by 5, divisible by 6, divisible by 7, divisible by 8, divisible by 9, divisible by 10, divisible by 11)

If after dividing a number x with y there is no remainder, we say that x is divisible by y. In other word, if the result of a division x/y is an integer, then x is divisible by y.
For Example: 51/17=3 . Hence, 51 is said to be divisible by 17.
But 9/2=4.5. So, 9 is not divisible by 2.

In mathematical problems we sometimes need to determine if a number is divisible by another. There are various tests to determine whether an integer is divisible by another integer without actually doing the division.

(a)  Is n divisible by 2?

 If the last digit of a number is 0, 2, 4, 6, or 8, it is divisible by 2. Stated otherwise, if the last digit is divisible by 2, the number is divisible by 2. Such numbers are also known as even numbers. e.g. 320, 136, 598,234, 572 are divisible by 2. But 131, 595, 233, 537 are not.

(b) Is a number (n) divisible by 3?
 If the sum of the digits of a number is divisible by 3, the number is also divisible by 3, The number is also divisible by 3. Let us test if 4321095 is divisible by 3 or not.
3+4+2+1+0+9+5 = 24. 24/3 = 8
Since 24 is divisible by 3, hence 4321095 is also divisible by 3.
(4321095/3=1440365)

(C) Is a number (n) divisible by 4?
We try to divide the number formed by the last two digits of n by 4. If that is divisible by 4, then n is divisible by 4. Lets say, we want to find out if 1536 is divisible by 4 or not. We take the last two digits and get 36. 36/ 4= 9. Hence we conclude that 1536 is divisible by 4. (1536/6 = 384)

(d) Is a number (n)  divisible by 5?
 Any number containing 0 or 5 as its last digit is divisible by 5. So, 3210. 5375, 320 etc. are divisible by 5.

(e) Is a number divisible by 6?
If a number is divisible by both 2 & 3, then it is also divisible by 6. For Example: Let us consider the number 2376 . Since the last digit is 6, it is divisible by 2. 2+3+7+6 =18, 18/3= 6 . So, 2376 is divisible by 6. (2376 / 6=396).

(f) Is a number divisible by 7 ?
If you double the last digit & subtract it from the rest of the number and the result is either 0 or divisible by 7, then the original number is also divisible by 7.
e.g 672, 2x2=4
67-4 = 63. 63/7 = 9
So, 672 is divisible by 7.

(g) Is a number divisible by 8?
The technique is similar to the one we use for testing divisibility with 4. |But now we use the last 3 digits to form a number and if that number is divisible by 8, the whole number is also divisible by 8 .

(h) Is a number divisible by 9?
This test is very similar to the test for 3, If the sum of all the digits of a number is divisible by 9, that number is also divisible by 9.
So, for 5321424 since 424/8 = 53, 5321424  is also divisible by 8.

(i) Is a number divisible by 10 ?
This test is very similar to the test for 3.You only have to check if the last digit is 0 or not. If it is 0, it is divisible by 10. Otherwise not. So, 100, 200, 120, 350 etc. are all divisible by 10.

(j) is a number divisible by 11?
 If we take the summation of every second digit and subtract all the other digits & the result is either 0 or divisible by 11, the number is also divisible by 11.
e.g 3223 is divisible by 11(3223/11=293) . this can be tested by the rule. 3+2-2-3=0.
Hence, it is divisible by 11.

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