PROPERTIES OF DIVISIBILITY

Property 1 :

When a number is divisible by another number, it is also divisible by the factors of the number.

For example,

18 is divisible by 6

Then, 18 is also divisible by 2 and 3 which are the factors of 6.  

Property 2 :

When a number is divisible by two or more co-prime numbers, it is also divisible by their product.

For example,

12 is divisible by both 2 and 3 which are co-primes

Then, 12 is also divisible by 6 which is the product of 2 and 3. 

Property 3 :

When a number is a factor of two given numbers, it is also a factor of their sum and difference.

For example,

2 is a factor of both 6 and 8.

Then,

2 is also a factor of 14 (= 6 + 8)

2 is also a factor of 2 (= 8 - 6)

Property 4 :

When a number is a factor of another number, it is also a factor of any multiple of that number.

For example,

4 is a factor of 12

Then, 

4 is also a factor of 24 (= 2 x 12)

4 is also a factor of 36 (= 3 x 12)

4 is also a factor of 48 (= 4 x 12)

Property 5 :

If an integer is divisible by two or more different numbers, then is it also divisible by the least common multiple of those numbers.

For example,

24 is divisible by both 2 and 3

Then,

24 is also divisible by 6 which is the least common multiple of 2 and 3. 

Practice Problems

Problem 1 :

How many numbers from 10 to 50 are exactly divisible by both 2 and 3 ?

(A)  10  (B)  8  (C)  14  (D)  11

Solution :

Numbers from 10 to 50 are 

10, 11, 12, 13, ..................50

Numbers which are divisible by both 2 and 3 are also divisible by the least common multiple of 2 and 3.

Least common multiple of 2 and 3 is 6.

Find the numbers from 10 to 50 which are divisible by 6.

They, are

6, 12, 18, ..........48

The above sequence is an arithmetic sequence.

The formula given below can be used to find the number of terms in arithmetic sequence.

n  =  [(l - a)/d ]  + 1

l = 48 (last term), a = 6 (first term), d = 6 (difference)

Then, 

n  =  [(48 - 6)/6] + 1

n  =  (42/6) + 1

n  =  7 + 1

n  =  8

Therefore, from 10 to 50, there are 8 numbers divisible by both 2 and 3. 

Problem 2 :

Check if 52563744 is divisible by 24.

Solution :

Decompose 24 into two factors such that they are co-primes. 

24  =  6 x 4   

24  =  8 x 3

So, 8 and 3 are the factors of 24. Moreover, 8 and 3 are co-primes. 

Check, whether 52563744 is divisible by 8.

In 52563744, the number formed by the last three digits is 744 which is divisible by 8.

According to the divisibility rule for 8, in a number, if the number formed by the last 3 digits is divisible by 8, then the number is divisible by 8.

So, 52563744 is divisible by 8.

Check, whether 52563744 is divisible by 3.

According to the divisibility rule for 3, if the sum of all the digits is divisible by 3 or a multiple of 3, then the number is divisible by 3. 

Add all the digits in the number 52563744.

5 + 2 + 5 + 6 + 3 + 7 + 4 + 4  =  36

The sum of the digits in the given number is 36 which is a multiple of 3. 

So, 52563744 is divisible by 3. 

Therefore, 52563744 is divisible by both 8 and 3 which are co-primes.  

By using property 2, 52563744 is divisible by 24 which is the product of 8 and 3.

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