Ever looked at a massive number like 4,839,276 and wondered, “Can I divide this by 2 without a remainder?” Before you reach for a calculator or start scratching your head over long division, there is a secret shortcut that mathematicians use to get the answer in less than a second. It’s called a Divisibility Rule.
In today’s post, we’re breaking down the simplest and most useful rule in the book: The Divisibility Rule for 2.
A natural number is divisible by 2 if the digit in the unit place (the very last digit) is an even number.
Specifically, you just need to check if the number ends in:
If it does, the entire number is divisible by 2. It doesn’t matter if the number is two digits long or twenty digits long—the last digit tells the whole story!
To see this rule in action, let’s compare two different numbers:
The Check: Look at the unit place. The last digit is 2.
The Result: Since 2 is in our “magic list” (0, 2, 4, 6, 8), 452 is divisible by 2.
Math Proof: $452 \div 2 = 226$ (A clean, whole number!)
The Check: Look at the unit place. The last digit is 3.
The Result: Since 3 is an odd number and not in our list, 953 is NOT divisible by 2.
Math Proof: $953 \div 2 = 476.5$ (We get a remainder!)
Learning these “shortcuts” isn’t just about finishing your homework faster (though that’s a great perk!). Mastering divisibility rules helps you:
Simplify Fractions: Quickly see if you can reduce a fraction by dividing the top and bottom by 2.
Prime Factorization: It’s the first step in breaking down large numbers.
Mental Math: It builds your “number sense,” making you more confident when working with figures in your head.
Test your new skills! Which of these numbers are divisible by 2?
A) 1,024
B) 5,771
C) 80
D) 9,999,996
(Answers: A, C, and D are all divisible by 2 because they end in 4, 0, and 6!)
The Rule: A natural number is divisible by 4 if the last two digits of the number form a number that is divisible by 4.
Example 1: 76532
Look at the last two digits: 32.
Since 32 is divisible by 4, 76532 is divisible by 4.
Example 2: 126
Look at the last two digits: 26.
Since 26 is not divisible by 4, 126 is not divisible by 4.
The Rule: A natural number is divisible by 5 if the digit at the unit place (the last digit) is either 0 or 5.
Example 1: 5785
The last digit is 5, so 5785 is divisible by 5.
Example 2: 6021
The last digit is 1, so 6021 is not divisible by 5.
The Rule: A natural number is divisible by 6 if it passes the test for both 2 and 3.
(Note: To be divisible by 2, it must be an even number ending in 0, 2, 4, 6, or 8).
Example 1: 7224
Test for 2: It ends in 4, so it is divisible by 2.
Test for 3: $7 + 2 + 2 + 4 = 15$. Since 15 is divisible by 3, the number is divisible by 3.
Result: Since it passes both, 7224 is divisible by 6.
Example 2: 124
Test for 2: It ends in 4, so it is divisible by 2.
Test for 3: $1 + 2 + 4 = 7$. Since 7 is not divisible by 3, it fails this test.
Result: It is not divisible by 6.
The rule for 7 is unique because it involves a bit of “trimming” and subtraction. There are two primary ways to check for divisibility:
To check if a number is divisible by 7:
Take the last digit (the units digit) and double it.
Subtract that result from the rest of the number.
If the result is 0 or a multiple of 7, the original number is divisible by 7.
Example: Is 861 divisible by 7?
The last digit is 1. Double it: $1 \times 2 = 2$.
The remaining part is 86.
Subtract: $86 – 2 = 84$.
Since $84$ is divisible by 7 ($7 \times 12 = 84$), 861 is divisible by 7.
For very large numbers, you can group the digits in sets of three from right to left and find the alternating sum/difference of these groups.
Example: Is 342,384 divisible by 7?
Group 1: 384
Group 2: 342
Subtract: $384 – 342 = 42$.
Since 42 is a multiple of 7 ($7 \times 6 = 42$), the entire number 342,384 is divisible by 7.
The rule for 8 is straightforward but requires looking at more than just the last digit.
The Rule: A natural number is divisible by 8 if the last three digits form a number that is divisible by 8.
Example: Is 93,624 divisible by 8?
Instead of dividing the whole number, we only look at the last three digits: 624.
$624 \div 8 = 78$
Because 624 is divisible by 8 with no remainder, we know that 93,624 is divisible by 8.
Pro Tip: If a number ends in 000, it is also automatically divisible by 8!
Building on our previous guide, let’s explore the shortcuts for three more essential numbers: 9, 10, and 11. These rules are incredibly useful for simplifying fractions and identifying patterns in larger datasets.
3. Divisibility Rule for 9
The rule for 9 is very similar to the rule for 3. You simply need to look at the sum of the digits.
The Rule: A natural number is divisible by 9 if the sum of all its digits is divisible by 9.
Example 1 (Divisible): For the number 6021, calculate $6 + 0 + 2 + 1 = 9$. Since 9 is divisible by 9, the original number is too.
Example 2 (Not Divisible): For the number 9005, calculate $9 + 0 + 0 + 5 = 14$. Since 14 is not divisible by 9, 9005 is not divisible by 9.
4. Divisibility Rule for 10
This is arguably the easiest rule to remember and apply at a glance.
The Rule: A natural number is divisible by 10 if its unit digit (the last digit) is 0.
Example 1: 902050 is divisible by 10 because it ends in 0.
Example 2: 9005 is not divisible by 10 because its unit digit is 5.
5. Divisibility Rule for 11
The rule for 11 is a bit more rhythmic, requiring you to alternate between positions.
The Rule: A natural number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places (counting from the right) is either 0 or divisible by 11.
Example: Is 1221 divisible by 11?
Digits at odd places (1st and 3rd from right): $1 + 2 = 3$.
Digits at even places (2nd and 4th from right): $2 + 1 = 3$.
Find the difference: $3 – 3 = 0$.
Conclusion: Since the difference is 0, 1221 is divisible by 11.
1. What are Prime Numbers?
A Prime Number is a natural number greater than 1 that has exactly two divisors: 1 and itself.
In simpler terms, a prime number cannot be divided evenly by any other number. If you try to divide a prime number by anything other than 1 or its own value, you will end up with a remainder.
Examples: 2, 3, 5, 7, 11, 13, 17, 19…
Proof: * $2 = 1 \times 2$ (Only two factors)
$5 = 1 \times 5$ (Only two factors)
2. What are Composite Numbers?
A Composite Number is a natural number that has more than two divisors. Essentially, if a number is greater than 1 and it isn’t prime, it’s composite!
Composite numbers can be broken down into smaller whole numbers.
Examples: 4, 6, 8, 9, 10, 12, 14, 15…
Proof: Let’s look at the number 4.
$4 = 1 \times 4$
$4 = 2 \times 2$
Factors of 4: 1, 2, and 4. Since it has three factors, it is composite.
3. Key Facts to Remember
To master this topic, keep these four important rules in mind:
The Smallest Prime: The number 2 is the smallest prime number. It is also the only even prime number!
The Smallest Composite: The number 4 is the smallest composite number.
The Exception: The number 1 is neither prime nor composite. Because it only has one factor (itself), it doesn’t meet the requirement for prime (two factors) or composite (more than two factors).
Prime Density: There are exactly 25 prime numbers below 100.
Quick List: The 25 Primes Under 100
If you are looking to memorize or reference them, here they are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
1. What is a Perfect Number?
A Perfect Number is a unique type of number where the sum of all its divisors (factors) is equal to exactly twice the number itself.
Example: 6
Divisors of 6 are 1, 2, 3, and 6.
Sum: $1 + 2 + 3 + 6 = 12$.
Since $12 = 2 \times 6$, the number 6 is perfect.
Example: 28 is also a perfect number.
2. Coprimes (Relatively Prime)
Numbers are considered Coprimes or Relatively Prime if they share no common factors other than 1. It is important to note that the numbers themselves don’t have to be prime to be coprimes with each other.
Example: 4 and 5 are coprimes because their only common divisor is 1.
Example: 2 and 3 are coprimes.
3. Twin Primes
Twin Primes are pairs of prime numbers that have a difference of exactly 2.
Examples: (3, 5), (5, 7), (11, 13), and (41, 43).
Pro Tip: Every pair of twin primes is also a pair of coprimes.
4. Prime Factorization
Every composite number $N$ can be broken down into a specific “DNA” of prime numbers. This is called Prime Factorization.
According to the Fundamental Theorem of Arithmetic, every composite number can be expressed as a product of prime factors in one and only one way. This unique combination of prime numbers, when multiplied together, will always result in that specific composite number.
Summary Checklist
Prime Numbers under 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Perfect Number: Sum of factors = $2 \times$ the number.
Coprimes: Highest common factor is 1.
Twin Primes: Prime pairs like $(p, p+2)$.
1. Special Classifications of Numbers
Beyond simple factors, certain numbers share unique relationships or meet specific mathematical criteria.
Perfect Numbers
A Perfect Number is a number where the sum of all its divisors (factors) equals exactly twice the number itself.
Example: 6
Divisors: 1, 6, 2, and 3.
Calculation: $1 + 6 + 2 + 3 = 12$.
Since $12 = 2 \times 6$, it is perfect.
Other Examples: 28 is also a perfect number.
Coprimes (Relatively Prime)
Numbers are called Coprimes or Relatively Prime if their only common divisor is 1.
Examples: 4 and 5 are coprimes, as are 2 and 3.
Twin Primes
Twin Primes are pairs of prime numbers that differ from each other by exactly 2.
Examples: (3, 5), (5, 7), (11, 13), and (41, 43).
Note: All twin primes are also coprimes.
2. Prime Factorization: The “DNA” of Numbers
Every composite number $N$ can be expressed as a product of prime factors in exactly one way. This process is known as Prime Factorization.
Example: Prime Factorization of 24
$24 = 2 \times 12$
$12 = 2 \times 6$
$6 = 2 \times 3$
Final Result: $24 = 2 \times 2 \times 2 \times 3$
3. Standard Form of a Number
The Standard Form of any natural number $N$ (where $N > 1$) is a more organized way to write prime factorization using exponents.
The formula is represented as:
$$N = p_1^{n_1} \cdot p_2^{n_2} \cdot p_3^{n_3} \cdot \dots \cdot p_k^{n_k}$$Where $p_1, p_2, \dots$ are distinct primes and $n_1, n_2, \dots$ are whole numbers.
Example: Standard Form of 54
Break it down: $54 = 2 \times 27$.
Continue: $27 = 3 \times 9$, and $9 = 3 \times 3$.
Combine: $54 = 2 \times 3 \times 3 \times 3$.
Standard Form: $54 = 2^1 \times 3^3$.
Quick Reference: Primes Under 100
For your calculations, here are the prime numbers you will use most often:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.