Understanding Divisibility Test Rules with Examples (Class 6 Maths)
An Overview: What Are Divisibility Test Rules?
Divisibility rules are a quick shortcut way to understand whether a number is divisible by another. This method can be used to find out the possibility of divisibility before carrying out the lengthy division method.
Example: consider number 246, and in this case, we don’t need to divide right away. Instead, apply a mind-sparking rule such that it would turn into the effort-reducing tactic. In such scenarios, divisibility would be helpful, especially with brain-bending numbers.
Example: imagine you have 12 cookies; you want to share them equally among 3 friends. A divisibility rule is a secret code--a magic-like trick--that tells you if you can share perfectly without breaking any cookies.
The secret for 12 is to add its digits: 1 + 2 = 3. Since 3 is divisible by 3, therefore 12 is divisible by 3. Hence, 12 cookies can be shared or split among 3. Hence, sharing 12 cookies among 3 people is possible--it’s a trick for fair-sharing cookies.
Hence, we notice that the divisibility rule can be applied to many numbers. The best part is that it gets very useful when dealing with fractions or while dividing one number by another.
Table Of Contents:
- Why Do We Need Divisibility Test Rules?
- Simple Divisibility Tests Examples
- Divisibility Test Rules from 2 to 10: Method Explained
➤ Divisibility Rule for 2
➤ Divisibility Rule for 3
➤ Divisibility Rule for 4
➤ Divisibility Rule for 5
➤ Divisibility Rule for 6
➤ Divisibility Rule for 7
➤ Divisibility Rule for 8
➤ Divisibility Rule for 9
- How Are Divisibility Rules Useful in Real Life: Examples
- Banking
- Sharing and grouping
- Coding and data science
- Billing or splitting items
- Simplifying Fractions
- Numerical Problems with Answers Based on Divisibility Test Rules
- Conclusion
Why Do We Need Divisibility Test Rules?
These number test rules save valuable time as they simplify fraction-based problems or numbers involving division.
Importantly, they play an essential role in HCF and LCM-based problems. Hence, they are considered vital and helpful for advanced-level math calculations.
Additionally, they make math faster; easier as they serve as a math-division shortcut! In math, divisibility rules work as special tools in simplifying BODMAS or PEDMAS calculations involving division.
To simplify the fraction 84, the rule for 4 works on both digits of the number. That is, you can quickly simplify it to 21 as 84/4 gives 21--you-cracked-it!
Thus, you can make a fraction simpler and easier using divisibility test rules.
Simple Divisibility Tests Examples
Suppose we consider the number 3648, and so it is necessary to check if it's divisible by [2], [3], or [6].
For the number two: You will notice that the last digit is 8 (an even number) in the number 3648. In our case, [8] turns out to be divisible by [2], and so the given number is divisible by two.
For the number 3: add the digits within the number: [3]+[6] + [4] + [8] = 21. Hence, the number 21 ends up smoothly divisible by number three. Thus—without a flicker of doubt—it stands confirmed as wholly cut-through by the number three (multiples of 3, grouping logic, remainder zero)!
For the number [6]: If a number is divisible by both [2] and [3], then it’s divisible by [6]; so, yes again.
Thus, in all these cases, there is no need to do actual division (or it can be done only if needed). Simply put, do divisibility and then decide whether to do division.
# Divisibility Test Rules from 2 to 10: Method Explained
Divisibility Rule for 2
Method: If a number is divisible by 2, its last digit has to be even [0], [2], [4], [6], or [8].
Examples:
124 → Last digit is 4 → Yes, divisibility can be carried out.
357 → Last digit is 7 → No, divisibility can’t be carried out.
Divisibility Rule for 3
Method: Add all the digits. If the sum is divisible by 3, the number is divisible by 3.
Examples:
456 → 4+5+6 = 15 → 15 is divisible by 3 → Yes, divisibility can be carried out.
842 → 8+4+2 = 14 → Not divisible by 3 → No, divisibility can’t be carried out.
Divisibility Rule for 4
Method: Check the last two digits, or if it has 00 as the last two digits. If that number is divisible by 4, then the whole number is divisible.
Examples:
1,236 → Last two digits: [36] → [36] ÷ [4] = [9] → Yes, divisibility can be done.
825 → Last two digits: [25] → Not divisible by [4] → No, divisibility not possible.
Divisibility Rule for 5
Method: If a given number ends with the digit [0] or [5] (0/5); hence, then it has to be divisible by the number [5].
Examples:
245 → Ends with 5 → Yes, divisibility can be carried out.
317 → Ends with 7 → No, divisibility can’t be carried out.
Divisibility Rule for 6
Method: If a number is divisible by both [2] and [3], then it's divisible by [6].
Example 1: [444] → It can be noticed that the last digit is even {so [444] is divisible by [2]}, and when digits are added {4+4+4 = 12}. This means that the number is divisible by [3] too.
So, the number 444 is not divisible by 6.
Example 2: [740] → Last digit is even and hence, divisible by 2.
But, 7+4+0 = 11 → No, divisibility of 444 can’t be carried out with 3.
So, the number 740 is not divisible by 6.
Divisibility Rule for 7
Start by doubling the last digit; then subtract the result (doubled or remaining digits) from the rest of the number (or remaining digits). If the outcome is divisible by 7, then the original number will too be divisible by 7.
Examples:
Let us consider the number: 203 → So, start by doubling last digit: 3×2 = 6; 20 – 6 = 14 → 14 ÷ 7 = 2 → Yes
301 → 1×2 = 2; 30 – 2 = 28 → 28 ÷ 7 = 4 → Yes; hence, the number 203 will be divisible by 7.
Divisibility Rule for 8
In this case, start by checking if the last 3 digits form a number divisible by 8.
Examples:
7,112 → considering the last 3 digits = 112 → 112 ÷ 8 = 14 → Yes. So, the number 7,112 will be divisible by 8 (7112/8=889).
3,765 → considering the last 3 digits = 765 → 765 ÷ 8 = not exact → No. So, the number 3,765 will not be divisible by 8.
Divisibility Rule for 9
Add all digits. When the digits' total (like [3]+ [6] + [9]) breaks evenly (gets divided) by 9, the entire number follows suit. This means complete divisibility of the number happens by 9.
Examples:
- 891 → the sum of 891 is given by addition of 8+9+1 = 18 → 18 ÷ 9 = 2 → Yes. So, the number 891 will be divisible by 9.
- 725 → 7+2+5 = 14 → Not divisible → No; so, the number 891 will not be divisible by 9.
Divisibility Rule for 10
Any number that signs off (gets divided by 10) with zero--like 50, 130, or 4,320--walks right into the divisible-by-10 club.
Examples:
- Consider the number 890 → Ends in 0 → Yes; so the number 890 is divisible by 10.
- Let's take the number 647 → Ends in 7 (not 0) → No; so the number 647 is not divisible by 10.
How Are Divisibility Rules Useful in Real Life: Examples
Divisibility rules are useful in many ways in real life, and they are as follows:
- Banking: Divisibility test comes in very handy when it comes to splitting money for dividing the money among many customers.
- Sharing and grouping: The test of disability is useful again when it comes to dividing students into equal groups--all students can be put in some group.
- Coding and data science: Computers can be programmed or algorithms can be made so that divisibility tests can quickly check the numbers when it comes to division purposes.
- Billing or splitting items: When it comes to product selling, finance, or dealing with money, divisibility tests can be utilized for equally dividing costs or quantities (like 100 apples among 4 people).
- Simplifying Fractions: One of the best ways for simplifying fractions is by using divisibility tests, as it easily reduces fractions by checking common divisors.
In other words, divisibility test rules are not just for math class, but they can be applied everywhere and in daily life!
Numerical Problems with Answers Based on Divisibility Test Rules
Q1. Does 728, when divided by 2, give some remainder or is completely divisible?
Answer: 728 is divisible [like splitting the 728 by an even number] by 2. Think about it this way: with 728 cookies, you and a friend can split them--an even [two-by-two-friendly] split with zero leftovers. This is a quick-check-math-trick and an even-steven-sharing-game.
Q2. Does 483, when divided by 3, give some remainder or completely divisible?
Correct: its digit sum makes fifteen when added; thus, divisible [three-way-split-possible] by 3. Just add the parts: 4 added to 8 and then added to 3, which gives 15.
Example: 15 things/gifts can be shared among three friends! This is a sum-it-up-fast-trick and a three-friend-share-test.
Q3. Arjun has 240 apples (all crisp-red); he wants to pack them. He gets boxes, and they can take four; so, will any be left over?
Answer: The solutions of this hinge on the divisibility rule-- that is, evenly sharing with no remainder. So, using the rule: are the last two digits divisible [so no-items-left-over] by four? Yes, the number is 10 when (40/4) when divided; therefore, no apples remain-- no lone-standing apples.
Example: To understand this situation: think of that you have a big bag of marbles. These marbles have to be shared among four friends. Divisibility rule tests are a way to go when it comes to getting the answer.
That is, whether you can distribute among all friends the same number of marbles. These have to be done without having any awkward leftovers. It's all about fair shares for everyone, and divisibility stands the test.
Q4. Suppose that a class holds 135 students---eagerly curious ones. Here, a challenge arises, and that is, can the teacher make teams of nine?
Answer: Let's figure it out using the divisibility rule: the teacher has to sum the digits together first. Considering students [total=135] and the divisibility rules, all digits when added together, make nine {[1]+[3]+[5]}.
And, guess what, the added number nine is divisible by nine--equal groups and no extras. So, the answer is yes--the teacher can, indeed, make fifteen teams--well-formed groupings.
Conclusion
Thus, divisibility turns out to be a reliable math trick, especially while dividing numbers by each other. Best Part: instead of doing long division, all one has to do is just add up the digits of the big number.
If the sum of the numbers is divisible, it can be shared perfectly, and in our case, it was by nine. So divisibility is the big number division finding tool that can save time.
