## DIVISIBILITY RULES

Basically, divisibility rules are shortcuts for finding out whether numbers are exactly divisible without doing long division calculations. If I ask you to determine whether 34521 is divisible by 4, you will most likely use the long division approach to find out, which will be quite a time-consuming process especially when the number is huge. Check out the divisibility rules below to enhance your math skills.

A number is divisible by 2 if it ends with zero (0) or an even number (2, 4, 6, 8). The numbers 456, 9322, 7658 are divisible by 2 because they all end with an even number. Likewise, the numbers 660, 320, 1800 are divisible by 2 because they end with zero.

A number is divisible by 3 if the sum of its digits is divisible by 3. Obviously, we know that 9, 12, 15 are divisible by 3 because we have memorized the multiplication table. This will help us find out if other huge numbers are divisible by 3. So, going by the divisibility rule,

• 405 is divisible by 3 because 4+0+5 equals 9 (9 is a multiple of 3)
• 381 is divisible by 3 because 3+8+1 equals 12(12 is a multiple of 3)
• 829 is not divisible by 3 because 8+2+9 equals 19(19 is not a multiple of 3)

A number is divisible by 4 if the last two digits are a multiple of 4 or if the number ends with two zeros. Here, too, knowing the multiplication table of  4 will help us check for the divisibility of huge figures.

• 548 is divisible by 4 because the number formed by its last two digits, 48 is a multiple of 4. Same applies to the numbers 27616, 712 and 5432.
• 4500 is divisible by 4 because it ends with two zeros.
Also See:  Maths Library

This is quite easy and obvious. Numbers divisible by 5 end with either 5 or 0. Example 500, 235, 625, 90 etc.

To check if a number is divisible by 6, we have to do two things—we check if the number is divisible by 2 and by 3. This is so because the prime factors of 6 are 2 and 3 and it is excellently logical to capitalize on that fact to check if a number is divisible by 6. So, let’s try some examples for the following numbers 5106, 508, 5912 and 636.

• For 5106, let’s check if it is divisible by 2. By applying the divisibility rule for 2, we realize it is divisible by 2 since it ends with 6(an even number). Now, applying the divisibility rule for 3, we sum up the digits and check if the result is a multiple of 3. Let’s do that now. 5+1+0+6 equals 12(12 is a multiple of 3). It satisfies both conditions so therefore 5106 is divisible by 3.
• For 508, it ends with an even number so it is divisible by 2. But it isn’t divisible by 3 because 5+0+8 equals 13, which is not a multiple of 3. In conclusion, 508 is not divisible by 6. Note that per the divisibility rule for 6, the number has to be divisible by both 2 and 3 to be eligible. The number has to satisfy both conditions.

Divisibility by 7 is quite strange and it requires some effort to comprehend it well. It becomes easy once you get by its quirks. This is how it is done. Lets actually use it on 245.

• Drop and double the last digit and subtract it from the rest of the digits and check if the result is zero or a multiple of 7. Otherwise, continue till you have exhausted all the digits.
• Dropping the last digit, 5, we now have 24. Now we double the dropped number, 5, and subtract it from the rest of the digits. Doubling 5 gives us 10. Subtracting 10 from 24 gives us 14(14 is multiple of 7), which makes 245 divisible by 7. Remember that if the number is quite huge, say 439236, we will have to continue with the dropping, doubling and subtracting till we now have a two-digit number to draw conclusions. In fact, try on the number 439236 and share your solutions with me in the comments section. Try for the following numbers as well: 4823 and 234.

For divisibility rule on 8, the last three digits of the number must be divisible by 8.

For a number to be divisible by 9, the sum of its digits must be divisible by 9.

Numbers divisible by 10 must always end with zero.

A number is divisible by 11 if the digit, when alternately subtracted and added, gives zero or a result which is divisible by 11. For example, the number 121, 12+1 equals zero; therefore, 121 is divisible by 11. Also, for the number, 2728, 27+28 equals -11, which makes 2728 divisible by 11.

Hit upon the comment section below if you’re facing any challenges regarding any of the above rules or perhaps some contributions and corrections you wish to share.

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### 2 thoughts on “Divisibility Rules in Mathematics”

1. Interesting,
But 7 is a bit complicated

• A visual representation will be quite helpful. Let’s try that on 4823.

Step 1: Drop the last digit (3)
We now have 482 instead of 4823.

Step 2: Multiply the dropped digit by 2 which gives us (3*2=6)

Step 3: Subtract the result (6) from the rest of the figure (482)
Which gives us 482 – 6 = 476

We now have a new figure 476.
We apply the operations from the previous steps once more. So let’s go

Step 1: Drop the last digit (6). We will now have 47 instead of 476.

Step 2: Multiply the dropped digit by 2 and subtract the result from the 47 obtained after dropping the last digit.
So, 6*2 = 12
Now, 47 – 12 = 35.

We have got a new figure again, 35. At this point, no need to continue because we can clearly see that 35 is a multiple of 7, which makes our original figure, 4823, also a multiple of 7.

I hope this clarifies any confusions